Due to its decentralized nature, as long as the internet remains relevant and we are still able to access it bitcoin and other cryptocurrencies will still be available for use. Bitcoin could and would survive a nuclear war long as there's people who would still be able to use it of course, and as long as the internet, or at least parts of it are still intact. I don't think monetary system would collapse, in my opinion it would become stronger than ever because money would be the biggest currency of the war, alongside soldiers and military prowess. Countless wars are won because you have money, I don't think this would not be the case even if a nuclear war stuck the planet. People will still find ways to use fiat just like the bottle caps on Fallout.