Since you already did the Cuban Missile Crisis turning into war, why not another World War III timeline?
Here are a couple ideas:
1. The Soviets follow through on their assumption that NATO's Able Archer exercise in '83 is a cover for a preemptive strike on them and decide to launch their own preemptive strike first. http://en.wikipedia.org/wiki/Able_Archer_83
2. Stanislav Petrov is not on duty September 26, 1983 when a false alarm is given at his missile control center. Believing the alarm to be real, whoever replace Petrov informs his superiors and events lead to WWIII. http://en.wikipedia.org/wiki/1983_Soviet_nuclear_false_alarm_incident