Nuclear winter scenarios have largely been debunked. Several different events in the 20th century had atmospheric impact on the same scale as modeled in a full nuclear exchange and the effects were modest.

People have kind of mythologized nuclear weapons far beyond their reality. One of the arguments against the use of nukes in policy circles is that this mythology is useful. The limited adverse environmental consequences in practice if demonstrated will greatly lower the threshold for subsequent use.

It might set society back a century, but with substantial knowledge of what was lost. Extinction from nuclear weapons is not remotely plausible.

If the AI wanted to wipe humans out it would have to have a sizable fleet of capable robots, as keeping the lights on over time is not just about calling APIs.