I don't know why almost everyone on here is assuming that Nixon would blindly and pigheaded-ly charge into WWIII. A mix of Kennedy glorification and Nixon demonification if I had to guess. Lets remember that in Nixon's first term in office, he showed what could amount to a masterful control of foreign policy, and did more than any other President of the era to limit nuclear arms and tensions. I'm not convinced that Nixon in '60 would be any different, and its possible that either a successful Bay of Pigs goes through, nipping the issue in the bud, or if Castro does hold power, the situation is cleared without any stand-off. Personality issues aside, Nixon in any year was a competent man.
What we can't answer about this, though, is the degree to which Nixon's 8 years in exile tempered and changed his views, nor how he was effected by events that took place during the Kennedy and Johnson presidencies. It seems a plausible thesis to me that the Cuban Missile Crisis itself, and the close call it genuinely was, may have affected his views on nuclear war. Also, during the 1960s, the vast Soviet deficit in nuclear arms that existed at the time of the Crisis was replaced by, if not parity, an ability to inflict catastrophic damage on the US, a fact which forced a different approach to the Soviets and to arms control. In other words, it cannot be definitively stated that Richard Nixon as President in 1962 would have been the same man who took office in 1969.
What I do think is that Nixon would have been more likely to follow some of the military advice that Kennedy wisely rejected. That alone could result in a dramatically different outcome.