A nuclear war can only be fought to the point where nobody has the capacity to keep fighting. You can't actually obliterate the planet. The global climate would be altered but it wouldn't be close to the worst the earth has seen. Insects, plants, rodents, and humans will all almost certainly survive a nuclear war. Animals in the wild are largely unaffected by high levels of background radiation because they don't live long enough for that to be what kills them and they reproduce before that anyway.

Nuclear war would suck for people but life on earth and a lot of the life on it would be just fine.