Briefly at first perhaps as a show of force. Most people would not want to go full mutual destruction. There could initially be some strategic strikes that hit small cities as a show of force.
I think the escalation risk would be countries that feel they are on the way out, have a history of not caring about their people and in effect feel they have nothing to lose. failing nations or governments going out with a bang vs. a whimper so to speak. One hit or fallout too close to a bordering nation could escalate quickly.
Another possibility for a first strike could be a few dozen stratospheric high altitude balloons carrying an EMP that wipe out electronics for a several countries or continents at the same time. I do not know how people might respond to that. There are also nuclear missile submarines all around the world that would be protected from initial kinetic and EMP damage.
I recall some discussion in fandom of the dissonance between that vision and the public propaganda lines of "Mutually Assured Destruction" and "Nuclear Winter". It seems to be a religious kind of debate where facts aren't welcome: There have been multiple nuclear weapons detonated in the last 80 years. Many still insist that even a single uncontrolled fission reaction is the equivalent of the end of the world.
I could see it starting that way. One side gambles that they can nuke one city and the enemy will deescalate to avoid a full exchange. But the other side is compelled to respond in kind, knowing that they can't sit and do nothing, and hoping that that a one-for-one exchange will be the end of it. And it might be. But if not, then it must be an all-out exchange imo.
Think about it like this: someone punches you. You punch back. If they do nothing, then that's the end of it (for now). But if they do it again - are you going to stand there all day exchanging punches? You now know that they're determined to keep hitting you. Your best chance is to throw everything you have at them right now. That minimizes the damage to you. That's clearly true if you're stronger than them. But if you think about, it it's equally true if you're weaker.
FWIW, I think everyone should stop worrying about nuclear war. The US would win hands down and everyone knows it.