This used to worry me too, but I read this article [1] and it cheered me up! Sure, most of civilisation would likely crumble if centres of production and distribution were nuked and vast tracts of the world would be irradiated, but it's by no means an ELE.

1: https://www.quora.com/In-a-total-nuclear-exchange-where-the-...

The impact which ended the era of the dinosaurs had an estimated energy greater than all the nuclear weapons in the world. Granted, it was concentrated in one area. But the result was thought to have effected the entire planet, with the sun being blocked out for several years, resulting in the collapse of plant life and drop in temperatures. The forests around the world may have also ignited after ejecta from the impact fell back to earth, temporarily superheating the atmosphere.

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.

I remember reading that some insects are more tolerant of radiation compared to humans.