The thing that would end our species due to nuclear war is the exact same thing that would end us from climate change - a collapse of the modern systems of society that we depend on for survival. A nuclear exchange won't set every human on fire, but it's the collapse of food production, healthcare, logistics chains, security, and eventual disease that follows. And because of the size of our population and how depended the majority of us are on these systems, it would likely happen extremely quickly before plateauing out to very small, scattered population groups that would struggle in a hostile environment.

Outside of extreme feedback loop scenarios, there is no way for climate change to disrupt food anywhere near the level of nuclear winter, and there would be no mass destruction of supply chains either.

So you agree they would survive