there will be ridiculous civil unrest, incomprehensible numbers will die

If things get desperate enough direct conflict between nuclear powers of basic resources like food and water could be a thing as well. Sure, life will survive. Microbes for sure. Abyssal creatures probably won't notice too much of a difference if they aren't oxygen starved before then (not sure about how much of a risk that is)? Some species or plants and animals are hardy enough to survive and some will get lucky.