>>12382287You would almost definitely have some kind of deficiency by the end of it, but you'd probably still be alive. The real issue as I see it would be storing or gathering enough material to grow that many mushrooms on. You get roughly 80% of the dry weight of your wood (or other suitable substrate) in wet mushrooms (90% is water) and about 10% as composted substrate that is suitable for secondary saprophytes.
Another issue is the energy required for the sterilization of your substrate. Obviously you can't get it from the grid, and nuclear winter implies no sunlight so solar is out, but wind might work. One simple solution would be to use a composted straw formula which can be used to pasteurize itself, but straw is more bulky than wood. Another solution would be to stock up on more wood that you could burn to sterilize the substrate. You might be able to use hydrogen peroxide to disinfect your substrate instead of sterilizing it, but natural substrates contain enzymes that rapidly decompose hydrogen peroxide so you'd need a fuckton to properly disinfect the substrate and it wouldn't provide any lasting protection. You could side step that issue by stocking up on wood fuel pellets, which have been heat treated enough to denature those enzymes, and similarly heat treated or inorganic supplements. Hydrogen peroxide drops in concentration over time, so eventually the bottles will just contain water, but the concentration is pretty easy to test so you could compensate by adding more until then. One downside is that once you're out you're out. There's no economical natural source and it's not really reasonable to expect to be able to synthesize it at home.
Thinking on it more, you might be able to get all the nutrients you need if you also stocked up on a bunch of extra seeds and few them as microgreens.