Many people have speculated about President Kennedy in such ways. What if he had not been in Dallas that day? What if he and his brother had not been assassinated? I propose a similar speculation. What if Lee Harvey Oswald had still assassinated the 35th President of the United States and JFK had lived to a ripe old age? You might argue that it almost happened that way because he almost wasn't president.
We often forget how close the election of 1960 was. It was one of those elections that literally could have had a different outcome if the weather had been different. In 1960, Kennedy prevailed with a margin of 0.17% of the popular vote over Nixon. Rumors were rampant about political machines and voter fraud in Illinois and Texas, two states which would have given Nixon the presidency had they gone his way. But unlike Al Gore, Nixon decided against contesting any state results, saying, “our country cannot afford the agony of a constitutional crisis.” Ironically, we later see that same value rise to the surface at his resignation.
So Nixon wins in 1960 . . . how does history change? Well, Kennedy lives, of course. He may have run for president again later and won as Nixon did. But I suspect that with his health problems he serves a term or two more as a senator from Massachusetts. Then, he hands the job over to one of his younger brothers--probably Robert, unless he has gone on to bigger and better things. So JFK writes books and gives lectures, fading into the sunset as an elder statesman.
Nixon becomes one of our youngest presidents, bridging the gap between WWII generation and the prosperous Eisenhower years and the up-and-coming generation of Americans looking toward a bright future. Both Kennedy and Nixon ran on very similar platforms, so policies may not have been so far apart. But surely things would have been different. Who knows how the Cuban missile crisis would have been different. Certainly Nixon was no less anti-Soviet than Kennedy.
Kennedy's challenge to go to the moon was never made. But we still have Sputnik and space technology beginning, so surely there is some kind of space race. Both Kennedy and Nixon were pro-civil rights, so there's some advancement on that front (probably mostly as a memorial after Nixon's assassination). Nixon keeps the pledge to appoint at least one black to his cabinet. But since the only segregationists were Democrats and there was no need later need for a "southern strategy" in 1968, African Americans become deeply attached to the party of Lincoln while the Democrats are split over segregation for a decade.
The biggest difference in this alternative history is that instead of LBJ, Vice-President Henry Cabot Lodge, Jr. is sworn in as president after the assassination of Nixon. Since the Great Society is so identified with President Johnson, it doesn't come to pass even though some elements of it may have later been enacted. Ironically, Kennedy had defeated Lodge for the Massachusetts senate seat in 1952. So perhaps like Nixon, JFK would have run for president again against Lodge in 1968.
President Lodge was a diplomat and perhaps this change in personality and temperament would have made a significant difference in regards to Vietnam, but maybe not. Ironically, Lodge later served as ambassador to Vietnam under Presidents Kennedy and Johnson, so he was already deeply involved in our own historical reality.
As for Nixon, we forget how popular he actually was because of Watergate. Nixon came back from political death to be elected president in 1968 and then was re-elected with the fourth largest margin in the popular vote and his total in the electoral college was only barely surpassed by Reagan's 49 state win in 1984.
The young first family has a romance with popular culture remembered as "Camelot" from the popular Broadway musical of the time. It is a much younger Elvis Presley who meets Nixon in the Oval Office. Perhaps even Marylin Monroe sings him "Happy Birthday" one year (but that's as close as she gets to the president of the United States).
With Oswald's assassin bullet in 1963, Nixon is immortalized along with his dream of a better America for all Americans. No one remembers him as "tricky Dick," but as a young president (perhaps our greatest) who inspires Americans for generations to come.
Don't let it be forgot
That once there was a spot
For one brief shining moment
That was known as Camelot.