Ralph Merkle’s latest article states: “If a nuclear war destroyed half our planet, [Bitcoin] would live on, uncorrupted. It would continue to offer its services… The only way to shut it down is to kill all the servers that host it. Which is difficult, because many servers host it, in many countries, and many people want to use it.”
If half of our planet get destroyed it means the other half will need to have steady energy source to power mining process and it will also mean the human who are survivors on the half planet must serve as nodes for the network to function. Once there is an explosion people will panic and saving a server might be the last thought that will come to peoples minds. If Bitcoin must survive a Nuclear war it relies on so many factors and the basics will be energy (power source) and the web
As for Bitcoin living on i won't doubt that but while it still lives it will get slower for transactions to get verified, cost of mining will multiply drastically, many nodes might forcefully shot down due to the attack and many will lose access to their wallet as explosion will destroy lots of valuables and documents. A big chance for the internet to shutdown completely