Your scenario seems pretty improbable, especially in today's world of published scholarship. If a claimed result for an important problem (worthy of Nobel consideration) were submitted it would receive tremendous scrutiny. If it was rejected, then that wasn't a random or accidental outcome. There was some flaw.
The later paper would also receive tremendous scrutiny and if published, no one found an error or hole in it. So, it is most likely sound. Not necessary, of course, but likely.
The new paper is published. Time passes. More time passes. There is a lot of scrutiny. Much of it is in public, not just private letters of a few academics. The result is eventually seen as so important that the Nobel committee awards the prize. It is extremely unlikely that it would ever be withdrawn if someone else claims priority. "Where have they been all these years?"
In mathematics, for example, proposing a theorem but not having a proof, or providing an incorrect proof is worth very nearly - nothing, though it might spark some inquiry that results in later success.
The Higgs Boson story is instructive. Higgs and Englert won a Nobel, but there was actually prior art on some of it.
I don't know of any field in which there is a formal mechanism for claiming priority after some other work is accepted.
That said, some fields are "hot" and there is a lot of parallel research. All the researchers are working with the same base of knowledge (papers, etc). Priority is almost a random situation.
I know of one example in CS in which two students solved exactly the same problem in the same way at the same time at different (high prestige) universities. It took the community a year to determine that both should get their degrees and share the honor on an important problem in language theory. But the process was very ad-hoc. The underlying reason for the simultaneous result was that there was a lot of interest at the time in answering that question.