Contrary to popular opinion, there is no single foundation for mathematics. Probably you're referring to ZF or ZFC, but most mathematics can be developed on the basis of axioms that are logically much weaker than that. If an inconsistency in ZF were discovered, we would analyze the inconsistency and then scale back to some weaker system that would avoid the inconsistency yet still suffice for 99%+ of mathematics. Much of the work of finding other candidates for foundations, and figuring out how much mathematics can be developed from them, has already been done by those working in the field known as "reverse mathematics." The basis basic text in this field is Simpson's Subsystems of Second-Order Arithmetic, but there is a growing literature.
We've already seen a dry run of this kind of instantaneous damage control. When Kunen's inconsistency theorem showed that Reinhardt cardinals were inconsistent, his work was hailed as a major achievement, but all we did was toss out Reinhardt cardinals and restrict ourselves to large cardinals below that bound.
For most mathematicians, "ZFC" is just an arbitrary trigraph that is cited when the need arises to specify a particular foundation for mathematics. I daresay many people who toss the trigraph around couldn't even state all the axioms of ZFC precisely. If we scale back to some other system that goes by some other trigraph, it won't take much retraining to learn the new trigraph. For most researchers, that will be the only impact on their day-to-day work.
