In a shocking subversion of standard FSTDT etiquette, a forumite steps forward and... deals with the OP. Some asspulling may occur.
A Nixon win in 1960 would've probably meant Watergate in 1964. We can safely assume that the Cuban Missile Crisis would play out in a similar manner, because a) there's only one sensible option in that scenario, and JFK took it, and b) this would be a very short post from anyone otherwise.
Watergate '64* would happen because Nixon was the kind of crazy bastard who would've done it anyway. Let's not forget that he was easily on course to win in 1972, but decided to cheat anyhow.
With Nixon resigning around 1966, this makes Henry Cabot Lodge the new Gerald Ford. By some theories, Lodge cost Nixon the election by pledging, unauthorised, to include at least one African-American in the Cabinet, thus losing votes in the South, so... the Civil Rights Act, or some form of it, might have passed around 1967, but similarly it might have taken a Democrat President from 1968 to pass it.
My guess is that Vietnam and the Civil Rights Act would have still happened, but in the longer term, the Reagan Presidency might have got lost in the shuffle. If Lodge passed the CRA, then the modern perceptions of the two parties might have swapped. That said, some form of Reaganomics, or else a Libertarian platform of some kind, would most likely have arisen around 1976, due to the economic issues brought about by the oil shock of 1973 and Watergate causing a loss of trust in government. The Civil Rights era and the Keynesian consensus might, if anything, have died a little earlier.
*is misleadingly named, because the tour went to Australia and NZ in 1965.