I think ABCBitcoin's answer about brainwallets for refugees is the best one until now.
I don't think Bitcoin will stop working and probably not even the hashrate would suffer that much. The exception would be a situation like pre-2021 when really 80% of mining were concentrated in a single country, and of course if just this country is under attack then the mining facilities could become targets. It would however perhaps be more lucrative to try to gain access to the miners' wallets via cyberattacks or intelligence work than simply destroy your adversaries' mining farms.
While I don't think that banks will be destroyed by a world war, what could happen is indeed a hyperinflation in one or several of the participating countries, leading to a currency reform like in Germany in 1948. If such a situation can be predicted, it's even possible that Bitcoin's price could benefit, as would gold's price.
International commerce in times of currency volatility due to economic warfare is another use case. And of course if your country is occupied, you could use Bitcoin (or Monero) to support movements and groups fighting for freedom, when using traditional banking has become too dangerous because the occupying country strictly monitors all local banks.
Of course this post considers that a "WW3" would be comparable in scale with WW1 and WW2. In both wars there were severe destructions but 80-90% of the world's population were almost unaffected by the warfare itself. WW2 for example was devastating in some parts of Europe and Asia but only punctually there was warfare in other areas, e.g. in Abyssinia, a small part of Northern Africa and Hawaii, but the US and India for example were barely affected. There are of course more apocalyptic scenarios where Bitcoin could take damage. However even in the case of a single 51% attack this wouldn't mean the end of Bitcoin; if it's only possible during an extreme situation then it may be possible to simply start the chain after the war (or when the situation improves) from a pre-attack block height. Even reverting the attack isn't impossible, Ethereum did that already once and they're still alive
