According to my quite likely wildly oversimplified understanding, a revolution occurred in the foundations of mathematics when Cantor's formulation of set theory was found to be inconsistent due to Russell's paradox, which ultimately lead to the development of Zermelo-Fraenkel set theory, thus putting mathematics back on solid formal foundations.

My question is, how much of a revolution did this process cause in the rest of mathematics, outside of the formal foundations?

I can imagine two extremes, with the truth presumably lying somewhere in between. On the one hand one could imagine that a crack in the foundations would break the whole of mathematics, with most theorems even in quite applied topics needing to be re-derived along quite different lines within the new system, in a process that would take many years. On the other hand I can imagine it not really making much difference at all, with most of the higher-level results being somehow independent of the low-level stuff below them, so that the old foundations could be swapped out and new ones put in without disturbing the structures that were built on top of them.

If I had to guess, I would say it was closer to the latter, since when reasoning about higher or applied mathematics we rarely have to go right down to the axioms of set theory. But I would appreciate knowing historically how it played out.