The point to remember in all of this is that Hermione, not Harry, is the primary time traveler. Harry is the tagalong. He goes with Hermione if he is there, but she is the one who must make the trip. She has the time turner.
With this point established, we may look at our first run through history: what happens if Hermione does not travel to the past? In order to be clear in each timeline, the Hermione for whom this is the first time through these few hours will be called the Original Hermione, and the one who has used the time turner to come from the future will be denoted as the Duplicate Hermione, because she is a temporal duplicate of herself.
The first point is that Buckbeak is executed. Our trio of troublemakers, Harry, Ron, and the Original Hermione don't actually see this, but they hear the axe fall and assume, correctly, that it is the neck of the hippogriff that is severed by it.
This creates the obvious problem, which any of my regular readers will spot immediately: Hermione is going to travel to the past specifically to save Buckbeak, and is going to succeed; but if she succeeds, surely she will know that Buckbeak was not executed, and so will not know that she needs to save him. However, the picture painted by the film suggests that the Original Hermione and companions who are leaving Hagrid's are unaware that Buckbeak has been saved by their later counterparts, and since McNair vents his frustration on one of Hagrid's beautifully big pumpkins, they hear essentially the same sound, or near enough that they do not realize the beast is still alive. Dumbledore, of course, knows that Buckbeak died, and will know that he was saved, but in classic Dumbledore wisdom, the wizard seems already to know what Hermione is going to do. Thus this will not be a problem.
What is more of a problem is that, moments before in the viewed history, the Duplicate Hermione hurls two stones or clods of dirt through an open window, breaking a pot and hitting Harry in the back of the head, to alert the Original trio to the fact that the executioner is coming. It is thought to be important to their situation that they not get caught at Hagrid's cottage. However, the warning does not seem to make any difference anyway, as they don't really scramble to leave the cottage until after Cornelius Fudge knocks on the door. Thus we have a throwaway, an action that accomplishes nothing but give us a bit of a time game: Hermione thinks to throw the stone because she sees the stone in the garden and recognizes it as the one she picked up in the cottage. The stones mean nothing.
Once the group leaves the cottage, Sirius Black, as a dog, spots them in the yard and tackles Ron, while Professor Remus Lupin is looking at the map so that he sees Peter Pettigrew (in Ron's hands) and follows, and does so at a moment when Professor Severus Snape sees him and follows in turn. Those events play out much as in the film; the duplicates never venture outside the grounds, and so essentially await the return of the originals.
It is at that moment that the first serious difference occurs: Harry is torn to bits by the werewolf Remus Lupin. Sirius attempts to protect him, but is badly battered in the process. Hermione has nothing to use against a werewolf, and cannot save Harry at that moment. The best--perhaps the only--thing she can do is help Snape get Ron to the castle.
Sirius winds up down by the lake, where the Dementors find him and destroy him, sucking out his soul and leaving his body behind. It has been a bad day, with Buckbeak beheaded, Harry hashed, and Sirius suctioned.
This puts Hermione in Madam Pomfrey's treatment room, possibly being treated for her own minor injuries, at the right time for Dumbledore to recommend to her that she can save more than one life if she goes back three turns. She does so, alone, probably expecting to save Harry but not knowing quite how to do it. This concludes what we call the AB timeline, the original history, the way events must have played before anyone left from the future and arrived in the past.
Before we consider the changes Hermione will have to make to create the second, CD, timeline, we should recognize that Hermione has been doing this all year, and that the book and the movie have both overlooked a minor point in this connection. Also, there is not a great deal of information available to tell us what has been happening. Hermione was not given the time turner to help Harry; she was given the time turner so that she could attend her incredibly overloaded course schedule, which included classes in different rooms at the same time. It is not clear how many such classes she had, or how long each class is; however, if we assume that she had to add two hours to each school day, five days per week, we have her working twenty-six hour days during the week and picking up an extra ten hours per week. Thus every five weeks she has gained two days on her peers. If school runs from early September to late May with a single break at Christmas for two weeks, we have almost thirty-five weeks, and Hermione has picked up almost fourteen days, two extra weeks of aging. This may be a conservative estimate. After all, with the extra classes she may want extra study time, and with the longer days she may need longer nights to catch up her sleep (how convenient to be able to sleep in and make the first class on time). It's not going to be a noticeable amount even if we stretch it to two months, but Hermione will be aging faster than everyone else.
This also means she is creating a series of anomalies--an original history in which she attends one class but not the other, and then an altered history in which she attends both classes. Any one of these in theory could go wrong; however, her intentions are quite clear, and should work. After all, it is not in this case that having missed a class she decides after the fact to fix that by traveling back in time to attend it instead (something she specifically determines not to do after Ron and Harry note her absence from a class), but rather that before attending either class she has decided to attend one and then travel back in time to attend the other as well. This being her intention, she should be able to maintain consistent N-jump outcomes through the school year, as long as nothing happens to derail the program completely.
With Dumbledore's words that she might save more than one life if she goes back three turns, Duplicate Hermione arrives in the early evening. Just as she does when Harry is with her, she works out that the first life to save is Buckbeak's, and she moves around behind Hagrid's house unnoticed. Her original self is still inside, and she wants to move her companions on their way, so she throws the rocks and manages to break the pot and hit Harry. This makes no difference, however, as they do not move until the approaching group knocks on the door anyway.
It is possible that Original Hermione might see Duplicate Hermione; but Original Hermione is aware that she can travel in time, and has enough on her mind at the moment that she'll probably not give it a second thought until later. That action replays in each subsequent history, as Duplicate Hermione steps out from cover to see her own hair, and Original Hermione gets a glimpse of herself ducking back under cover.
She has to save Buckbeak without Harry's help; but since she has no notion of having Harry's help anyway, she will do this. She will of course bow, as she was taught, and then probably use the dead ferrets in combination with the chain to lure the hippogriff into the forest out of sight.
Having saved Buckbeak, she now moves deeper into the forest and circles around to a spot from which she can more easily observe the Womping Willow, the spot from which Harry will emerge. Eventually he does so, and Remus, struck by the moonlight, transforms into the werewolf. This Duplicate Hermione has probably already considered her options, and so without hesitation she makes the wolf call--and then realizes that she has lured a werewolf directly toward her. She runs from this, hiding among the trees, and is saved by Buckbeak, who drives Lupin away from her.
However, Hermione is unaware that Harry, whom she has just saved from the werewolf, has run off after Sirius and is about to be killed by the dementors. By the time she knows this, it will be too late; Harry and Sirius will be dead. She will return to the castle somewhat dejected, but before she returns to the medical ward her Original self will depart, creating yet another altered history. The first altered history, the CD timeline, comes to a less than glorious end.
The new Duplicate Hermione does not know that Harry was saved from Lupin by the wolf call from a previous duplicate Hermione; she only knows that she has to figure out how Harry got down to the lake, and somehow save his life. She will again realize that she has to save Buckbeak, and will do so much as described in the CD timeline. Then she will move to that spot from which she can effectively watch the Womping Willow, and see the Original group emerge. Seeing Harry confront Lupin, she will realize that he is about to be shredded, and will make the wolf call. After that, she flees from the werewolf and is rescued by Buckbeak.
Harry runs toward the lake to catch up with Sirius. Duplicate Hermione, however, now knows that Harry will die there, and once she is free of the werewolf she heads for the lake, emerging on the far side as she does in the film, but without Harry.
That is not what happened in the book. In the book, Harry goes to the far side of the shore by himself, trying to find his father, whom he believed cast the Patronus that saved him. That is why the book fails but the movie succeeds. In the movie, Harry and Hermione race to the shore together to seek to save Sirius; thus it is reasonable to suppose that when alone, Hermione also will race to the shore opposite Original Harry and Sirius, to attempt to save Harry from the dementors closing on them. This is the critical moment at which you have to believe something for which there is almost no evidence, or the film fails: Hermione must successfully protect Harry from the dementors, and she must do so from the opposite side of the lake.
That's not as entirely impossible as one might think. First, she has seen Harry cast the Patronus before--when Malfoy faked being a dementor at a Quidditch match to scare him. Besides, she is the smartest witch of her generation, so attempting to cast a spell she has seen done exactly once is not completely impossible. It has a non-verbal non-physical component, that the caster must think of a happy thought; but the ridikulus spell used against boggarts also has a non-verbal non-physical component, so it is possible that she might pick up the similarity here. Also, the real difficulty in most castings of the Patronus is that the caster must think something happy while having all happiness sucked out of him by a dementor. Hermione is across the lake, not subject to the dementors' effects, and so should find the spell easier in that context than it would be were she in the midst of the trouble. In any event, she must cast a patronus from the opposite side of the lake, and it must save Harry's life. Note that it does not have to do more than that, and given her determination to save Harry she might succeed in getting the dementors to leave him but take their targeted Sirius. After all, the dementors weren't really after Harry, and if they get Sirius they might be driven from Harry fairly easily.
It is not clear what Duplicate Hermione will do next. Probably she will remove the chain from Buckbeak and release him into the forest to fend for himself. There is no saving Sirius, because the dementors have already done their worst. She has to be back to the door outside medical before the clock stops striking, so probably she will be walking up the corridor toward a smiling Dumbledore with news that Harry is alive--news Dumbledore already knows, and which she will never deliver, because Original Hermione is now standing in Madame Pomfrey's rooms with Original Harry, turning the time turner to go back to save Sirius and Buckbeak. The EF timeline ends, and a new history is about to be created.
This time, Duplicate Hermione's information is different, and she has a companion, and it impacts how the new history plays.
As mentioned, Duplicate Hermione wonders, as she does in the film, why the Original trio are not leaving Hagrid's cottage. She then sees the stone that broke the pot, and also must recall that Harry was hit by some unexplained object, and thus that she must have thrown a stone or dirt clod at him. As mentioned, this is a wasted action, as the trio does not leave until the executioner is at the door anyway. She has Harry's help this time, and two working together move Buckbeak. They are, of course, unaware that Hermione was able to do this herself in a previous history.
Most of the events from there play out as we see in the film, with one exception. They circle around, using Hermione's wolf call to lure Lupin away from Harry, then flee through the forest, rescued by Buckbeak's intervention, then, because Harry wants to save Sirius, they rush out onto the far shore of the lake, where they see Original Harry and Sirius, and the approaching dementors.
Here, Duplicate Harry knows only that someone cast a patronus of some sort, and it saved him. It is likely that this patronus was not well-formed, and was not terribly powerful, but it did save him. He might be looking for who did it; however, he did not see himself do it, so he does not think it was his father. He emerges into the light, and when he sees that there is no one else, it does not occur to him (or to her) that Hermione must have done it. Thus he does it himself, casting the patronus powerfully enough to save himself and Sirius.
While Sirius and Original Harry are being moved to their respective locations, Duplicate Harry and Duplicate Hermione are off to the astronomy tower aboard Buckbeak. Once Sirius is locked up in the tower, they release him, and he leaves on the back of the hippogriff while they race back to the medical wing before the loop ends. Again, though, they do not make it. The GH timeline is not the end; they have to play through this act one more time.
There is one difference between the timeline just outlined as the third change (the GH timeline) and the one we see in the movie, and that is this: Harry saw himself, and thought he was seeing his father. That means that the timeline in the movie is one generation further along the sawtooth snap. Harry could not have survived the original AB history because even had Lupin not torn him to shreds, the dementors would have gotten him. Therefore he could not have traveled back with Hermione to create the CD timeline, and could not have rescued himself, and could not have seen himself rescue himself. Hermione had to rescue him there, in the EF timeline. Thus for Harry to have seen himself, he must have traveled back with Hermione after she rescues him from the dementors in the EF timeline, making that the GH timeline in which he saw himself; but then when he travels back, he is a slightly different Harry, one who thinks he saw his father cast a patronus from across the lake, and thus is looking for his father when he finds his Original self there. This, in turn, means that it is the IJ timeline when Harry, standing on that shore, realizes that he saw himself, and casts the patronus with confidence, knowing that it was he who did so last time. His argument that he could do it easily because he knew he had already done it makes some sense in that context.
This resolves all the anomalies. Harry and Hermione rescue Sirius and send him away on Buckbeak, and return to the medical wing. Their originals are just leaving via time turner to do what they, the duplicates, have just finished doing, and Dumbledore admits them to the room as it is vacated. Time continues, and the history we see in the film is the only history anyone knows. The departing Original Harry and Hermione are exactly identical in every way to who the returning Duplicate Harry and Hermione were three turns back when they left, so they will repeat history exactly as we have seen it.
I said I might owe J. K. Rowling an apology. It turns out that I do not. The book version of the story fails; Harry dies, victim of the dementors by the side of the lake, with no one to save him, and so can never make the trip from the future to save himself. It happens that in trying to simplify events in the movie, raise the action, and give Harry someone to whom to express his thoughts in that critical scene, the filmmakers blundered into a solution to the anomaly that works. I'm pleased to be able to resolve the movie version, but regret that the book version is not so easily fixed.
Harry Potter fans who are reading this sooner rather than later might be interested in reading my MySpace Harry Potter blog. It attempts to predict the contents of the forthcoming final book (although for most readers that book will already be out, given its imminence as I write this). You'll find a summary and links to the detailed posts in Arithmancy: A Summonation, if you are a Potter fan.