Good old pencil and paper + carrier pigeons.
Yeah, ok, let's all start mining with pencil and paper. Realistically, it isn't going to happen, you could do maybe a couple hashes a day assuming you worked at it 24/7 and in a real survival situation nobody is going to have time to do that. Nobody has time to do that now, that's why computers do it for us.
Now, let me turn that question around: How can you survive without electricity? If the world loses electricity, you wouldn't be worried about bitcoin, you'd be worried about survival.
Seriously? You don't believe people are going to be able to survive without electricity, but you believe Bitcoin will survive without electricity because everyone will just mine with pencil and paper? How are all these dead people going to find time to do something that even living people today won't do?
As far as how I'd survive without electricity, I've done it before and I could easily do it again. I have, or can easily obtain or make, the four basic necessities of survival.. water, food, shelter and fire. I also have the means to defend those necessities against anyone who would try to take them from me. I don't need anything beyond that, everything else is a luxury. I would be (and have been) quite bored at times without some of the luxuries electricity provides for me, but I have no doubt I would survive.
We still have a few blacksmiths around, who could repair the generators with hammer and anvil.
All jokes aside, blacksmithing (or metalsmithing in general) is a good skill to learn and can be a very enjoyable and profitable hobby. I learned the basics of blacksmithing years ago from a friend who happened to work as a blacksmith at a local amusement park, and lately I've been getting into silversmithing and goldsmithing.. not because I need to, but because it's a fun way to relax and because I get the satisfaction of making something with my own hands.