OTL, given the association of Oswald with pro-Castro elements, there was a short-lived concern about a "communist" element in the assassination. IMHO had there been a provable link to a real communist group, this would have been very bad. In 1962 the view of communism was that it was a monolith directed from Moscow, everywhere and in every way. The Moscow-Beijing split was not yet obvious, the only "independent" communism was Yugoslavia and that was not universally accepted. In this environment Kennedy's assassination would be seen as directed by Moscow and as such would be a causus belli. In 1962, as has been explored here in many T/L's the USA had a significant superiority in nuclear weapons and delivery systems, and with the USA striking first with NORAD defenses on full alert it is probably very few nuclear weapons would explode on US soil, perhaps even zero. Soviet bombers needed to be forward staged, they had limited air to air refueling and the few ICBMs needed significant time to be fueled and made ready so a US first strike could very well chop off much of the Soviet nuclear force capable of hitting the USA. What happens in Europe, hard to say.
IMHO there is absolutely no way that the USSR could say "sure it was a communist group, but WE had nothing to do with it" and be believed. If the group was Castro sponsored, in retaliation for CIA attempts on Fidel, the best would be the Soviets throw him under the bus and the CStro regime/communism is out of business in Cuba. Under these circumstances the USA might, just might, go to war with the USSR. Of course relations between the US and the USSR would be in the deep freeze for a long time, and I expect the if the US gets in any proxy wars, like Vietnam if it happens, the US will take the gloves off (ground forces, bombing north & more) right away.