What a lovely question!
The first thing to ask is whether the story is internally consistent -- unlike a story in which John kills his grandfather before Grandfather fathers John's mother. That appears not to be a problem here; there's no obvious hint of an event having to have happened and not happened. Instead, we simply have surprising set of internally consistent occurrences.
On, then, to responsibility. Since we have a causal loop here, there's no clear way to say which of Peter or Kurt is causally responsible for the theorem's having come to be stated and proved. If pressed, we might say that each gets equal causal credit,m though your mileage might vary on the apportionment. Indeed, neither is the originator or initiator; the most that can be said is that both were crucial parts of the process through which the theorem came to be.
What of intellectual credit?
Well, Peter didn't think the ideas up on his own. He learned them by reading about them, having been taught them, working through the proof and what not. So he doesn't get any intellectual credit. But as this story goes, Gödel didn't think the thing up either. (To make the story as perplexing as possible, we should assume that Peter simply gave the whole shebang to Kurt, rather than just giving him a crucial bit.) And so Gödel gets no intellectual credit either. Somehow, the world just burped forth Gödel's theorem, praise be to the Lords of Time!
Wonder if it really happened that way?