What examples are you thinking of? I can think of only one case where this has been the case (Mochizuki and the ABC Conjecture), but that turned out to have so much fanfare precisely because mathematicians did not agree on what distinctions were being made and this was the only time in living memory that this had occurred (the general consensus is that Mochizuki's proof is simply too obfuscated to make heads or tails of). As such the ABC Conjecture is not considered solved.
However, that is almost always not the case, even on the cutting edge of mathematics and even when making other breakthrough discoveries (e.g. Fermat's Last Theorem).
> E.g. it would be like a psychologist asking how an results of a well proven result would be different if all participants wore red shoes.
And to be clear the major reason why fields like psychology are termed to have a "replication crisis" is because well-known results are being overturned, not just cutting-edge ones.
EDIT: I see now that OP also refers to the ABC Conjecture.