what if Boris Yeltsin was drunk or really terrified of the incident and actually took orders to fire nuclear weapons, boom civilization is kinda nearly dead?
In Russian Federation, the consent of the majority of 3 people is required to give the order for a nuclear strike: the President, the Chief of the General Staff and the Minister of Defense. Therefore it's flatly impossible for two of them be persuaded that one rogue rocket requires launch-on-warning revenge strike. Especially in 1995 when the US-Russian honeymoon was still in effect.
Most likely, as of 1995, even if it was an ASB-accidental launch of a real missile, the Russian army would first wait for a report of the detonation of a nuclear device inside the territory of the Russian Federation, and then the Russian government would request urgent communication by a red line with Washington.
One missile cannot ignite a full-scale nuclear war.