Of course, there will be power. It depends on the scale of a nuclear war. A few nukes won't pose an existential threat. Let's say 1000-2000 nukes are exchanged. That will destroy infrastructure. At that point, something like Bitcoin will be a complete luxury. I don't think you understand the impact an all-out nuclear war will have. People will be scrambling to find water, food, shelter, heat, etc. Plus will have to deal with radiation. Crops will die. It will be a major clusterfuck. Nobody will want to know how much Bitcoin you have - lmao! You'd better place your computers in a Faraday Cage. Russia / USA will detonate EMP devices before nuking each other. The EMP blasts will destroy electronics over vast areas - unless they are protected. Cars won't run - unless they are older types that don't have computers in them.
So sure, Bitcoin might live on - might. But I doubt it. Who's going to keep mining after a nuclear war?! Who's gonna trade / buy? Get real!
Don't assume that nuclear war will involve 2000 nukes. It could, but it also could be a smaller number and localized to specific countries and regions, causing absolute havoc from which we can, (and form which some people will) escape from.
If so, the ability to move (both one's person physically, and one's wealth) will be crucial. If you are somewhere where you are physically safe, you also want access to wealth. Banking system is likely to be severely disrupted. And there are only so many carrots / chickens etc we can barter with (if we even have access to them in the first place). Humans have always used mediums of exchange. Gold has its limitations (but also its uses). Bitcoin is globally accepted, globally transferrable, and infinitely divisible, and not controlled by any military regime. Whether more energy or less energy, the mining algorithm adjusts to supply and demand, meaning that there will always be a point at which some parties are incentized to mine. Bitcoin and Gold are of course not the only things you want to be holding in a time of crisis. We also might want knives, guns, long storage food, gas cookers, medical kits, matches, torches, blankets, etc. But Bitcoin, Gold, multiple passports, and cash in a range of currencies in small denominations, are right up there on the list in my "survival kit" (along with a network of good reliable true friends and communities arond the world).
