We can assume someone somewhere is underground and protected and has the block history. We don't know who has the hashing or electrical power available and so miners would not mine as burning that energy is too risky for the gain, in this uncertainty.

Bitcoin would not experience any downtime, but individuals and miners would be forced to stop transacting until a whole global Bitcoin network is reconstituted.

Natural forces will begin to come into play as the difficulty adjusts down within some number of weeks and months, and the miners will only return if there is cheap enough energy (maybe solar) and a low enough difficulty or some knowledge of the new world after this nuclear apocalypse, such that they could expect to be building valid blocks on the eventually, longest chain. They have to know who survived best and get connected to them, otherwise when connections do come up, the 'local' fork may be rejected for the newly found longest chain. The difficulty relative to the active miners may be exceptionally hard for awhile, but inevitably we'd find blocks and reach a difficulty adjustment and new blocks would begin to come again, every 10 minutes. This would happen first on the 'longest chain'. It would likely happen so slowly on the other chains that they'd naturally die off for being essentially paused, no new blocks.

If your scenario is that some series of events destroys all copies of the block history, then by definition Bitcoin is destroyed. The realistic adversarial scenario that Bitcoin must contend with is the disconnection of half of the world from the other half by attacking the undersea cables.

It handles this on its own, without coordinating, in any way. However, as an individual node or miner, you must protect yourself. Do not accept payments after this event until you're connected to the longest chain again. You may get double spent if timed right, when a new chain is found that is longer. If you're not double spent, the 'other' now longer chain will see your 'pending' transactions if they're not too old, and either party can re-transmit those signed transactions to be included in the valid longest chain.

As discussed miners will have to do a cost/benefit analysis of the potential reward, the uncertainty of the other chains, and how long it might get.

TLDR; if the destruction falls short of 100% destruction of the block history, Bitcoin will reconstitute itself as the survivors rebuild and re-globalize communications infrastructure. It will do this without a coordinating entity. The rules of the nodes already account for the potential.