Calling this an x-risk seems to be a case of either A) stretching the definition considerably, or B) being unduly credulous of the claims of political activists. A few points to consider:
1) During the height of the cold war, when there were about an order of magnitude more nuclear weapons deployed than is currently the case, US military (which had a vested interest in exaggerating the Soviet threat) put estimated casualties from a full-scale nuclear exchange at 30-40% of the US population. While certainly horrific, this falls far short of extinction. Granted, this was before anyone had thought of nuclear winter, but:
2) Stone age humanity survived the last ice age in Europe, where climate conditions were far worse than even the most pessimistic nuclear winter scenarios. It strains credibility to imagine that modern societies would do worse.
3) The nuclear winter concept was invented by peace activists looking for arguments to support the cause of nuclear disarmament, which a priori makes them about as credible as an anit-global-warming scientist funded by an oil company.
4) Nuclear winter doesn't look any better when you examine the data. The whole theory is based on computer models of complex atmospheric phenomena that have never been calibrated by comparing their results to real events. As anyone who's ever built a computer model of a complex system can tell you, such uncalibrated models are absolutely meaningless - they provide no information about anything but the biases of those who built them.
So really, the inclusion of nuclear war on x-risk lists is a matter of media manipulation trumping clear thought. We've all heard the 'nuclear war could wipe out humanity' meme repeated so often that we buy into it despite the fact that there has never been any good evidence for it.
Nuclear winter doesn't look any better when you examine the data.
Have you done this? I've asked random climate physics folk (pretty smart and cynical people) to take a look at the nuclear winter models, and they found them reasonable on the basic shape of the effect, although they couldn't vouch for the fine details of magnitude. So it doesn't look to me like just a narrow clique of activists pushing the idea of nuclear winter.
The FHI's mini advent calendar: counting down through the big five existential risks. The first one is an old favourite, forgotten but not gone: nuclear war.
Nuclear War
Current understanding: medium-high
Most worrying aspect: the missiles and bombs are already out there
It was a great fear during the fifties and sixties; but the weapons that could destroy our species lie dormant, not destroyed.
But nuclear weapons still remain the easiest method for our species to destroy itself. Recent modelling have confirmed the old idea of nuclear winter: soot rising from burning human cities destroyed by nuclear weapons could envelop the world in a dark cloud, disrupting agriculture and the food supplies, and causing mass starvation and death far beyond the areas directly hit. And a creeping proliferation has spread these weapons to smaller states in unstable areas of the world, increasing the probability that nuclear weapons could get used, leading to potential escalation. The risks are not new, and several times (the Cuban missile crisis, the Petrov incident) our species has been saved from annihilation by the slimmest of margins. And yet the risk seems to have slipped off the radar for many governments: emergency food and fuel reserves are diminishing, and we have few “refuges” designed to ensure that the human species could endure a major nuclear conflict.