I'm looking for the name of a Traveller adventure I read on the Internet long ago. It concerned the characters landing on an atomic-age world to rescue someone (the daughter of someone important?), or maybe just to trade. The important thing is, nuclear war broke out between the two superpowers on that planet and the players had to get out amid the ensuing chaos. Dice were thrown to check the escalation of the situation.
There are some differences, but Lee's Guide to Interstellar Adventure was a third party product for Traveller released by Gamelords Ltd in 1982 and contains 10 worlds, each with an associated adventure outline. "World 6: Back From the Ashes" on p23-26 is about a full-scale nuclear war (with bioweapons as a follow-up).
No dice-rolling for escalation though, and action is focussed on assisting the several hundred survivors on the two moonbases (one base from each superpower) who have just lost their planet.
However, World 10: War Zone in the same product is about players sneaking past an Imperial interdiction blockade on another balkanised planet where the equivalent of World War 2 is kicking off. Closer to what you're describing, except that the war doesn't go nuclear.
Lots of planets were nuked from space in the Rebellion and Hard Times and there were various adventures in which single illegal nuclear weapons were a focus, but this is the only adventure I remember where a balkanised world nuked itself until it glowed.
https://amber.zone/2014/08/30/amber-zone-one-second-to-midnight/ Author (likely pseudonym) Alegis Downport.
Two countries have engaged in a conventional war and one is losing so it is about to go nuclear. There is a daughter to rescue.
Bonus: Another 'nuclear related' scenario is Thunder on Zyra in JTAS #11 by Roger Moore. It is more or less a Merc ticket.