climate change was never going to kill all humans.

There's some scenarios where it could. One plausible one off the top of my head is: climate change makes the flow of the Indus (which has its headwaters in the glaciers of the Himalayas) less reliable. Two nuclear powers (India and Pakistan) are highly reliant on the Indus for agriculture. Things escalate out of control, and when nukes start flying it trips the terrifyingly ramshackle Dead Hand system that Russia's got hooked up to their nukes (it's not always on, but maybe this happens during a time of heightened political tension).