Robert L forward wrote an entire novel (Timemaster) exploring this, sort of, to show the variation of tie travel that might actually be allowed in the real universe.
In his story, you can change history, sort of. The universe requires that the result must be self-consistent. There can be no paradoxes (paradoxes are equivalent to breaking conservation of energy/momentum, which is not good). So, if you go back n time, you can kill your grandfather before he sires your parent, but someone with your father's genetic code *will* be born, by hook or by crook. To the protagonist, it looks like he always has free will, but to the reader, it comes across like destiny.
now for this part...
This is certainly one way of doing time travel. It's basically the self-correcting version of time travel. Which in my mind, assumes some conscious effort on the part of the TimeStream to decide what is important to retain/re-map so it happens, and what part is acceptable to deviate from. Usually such stories allow for minor changes, and in fact, the reality would have to accommodate at least one major change, primarily the injection of a new mass of a time traveler.
After all, if I travel back in time to a book depository and bump into some guy about to take a shot with a rifle, who's to say that shot is more important than the one where Dick Cheney shot that guy in the butt.
Or as I was crossing the street in Dallas 1964, this one dude thought the outfit I was wearing was really cool so he invented a new fashion of clothing. Why is one change more important than another to allow or block?
As just as valid a idea, the one I've seen incorporating quantum physics is that there's already alternate realities for every decision point (both sentient being induced and quantum dice roll of stuff that is decided "randomly" by physics) or that these alternate realities spawn into existance the moment a choice happens. That part's not too important because it's chicken and the egg to time travel. By the time you have time travel, you got chickens and eggs already.
Anyway, when you time travel and kill your grandpa, You have landed in a DIFFERENT reality than the one you came from. You now find yourself in a reality where it's version of your grandpa died by a mysterious stranger who appeared from nowhere. As such, YOUR actual grandpa still existed and you got born. No paradox, no drama. You could (tech permitting) return to your own reality where he's just fine, or travel forward to the future branch of this "dead grandpa" reality chain to see how things turned out, though technically, this would be a THIRD alternate reality as again, it is the one where you just popped in from nowhere to observe it, as opposed to the one where you did not appear.
There's probably issues with adding mass/energy for myself in the target reality either because I'm made of the same matter (from another reality) or because I really am adding dirt from Reality A to Reality B. That's something for folks like Umbran to know..
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Suffice it to say, I hate the "can't change anything" style of time travel as it's so much BS with my view of reality and inhibiting in a game where we are presumably featuring time travel (aka prophecy/actual time travel).
To bring this back to topic, for a game that has Prophecy/time travel, I favor an interpretation that enables player choice to matter in the sense of if they try to make a change (and if they are successful).
I do think the "implied responsibility" issue Umbran raised is very important not to dismiss. For PCs doing a spell for "what's going to happen tomorrow" or "is this course of action a good idea" it was the player's choice to seek that info, and thus their choice how to use it. If the GM foists a "you are the chosen one, this is your quest" or "the king will be murdered next week" kind of hand-me down prophecy, that's the danger zone as it pretty much defines the direction the game is going outside of player choice.
That said, even in that framing, it should be the player's choice to go along or disrupt the prophecy. That should be the very point of giving it to the players in that the GM assumes (hopefully by knowing his players' preference) that they would enjoy having some adventures about fulfilling or preventing the prophecy.
Within the prophecy about the king dying, how that's phrased and how it relates to the PC's also matters.
If it says "the PCs shall kill the king and save the kingdom" or some such, that doesn't mean it's a done deal. Dice have to be rolled, fights have to happen. But it does indicate that going to kill the king is a good idea and that it will result in a happy ending if they are successful. It may also be possible the players come up with an alternative, like arresting the king, or even talking him into being a good guy. Which of course invalidates the prophecy but still might be a good outcome.
Since that prophecy implies player action, you have to plan for players rejecting that action (doing something in opposition) or accepting that recommendation.