IOTL he only missed it by 15 votes on the second ballot. Let's say he receives the nomination. Does that line JFK up even more perfectly up for the '60 nomination than OTL or does a backlash develop?
I'd say that Nixon's election in 1960 becomes essentially assured. Chances are the Democrats would be faced with nominating either Johnson or Humphrey. It's debatable at best whether a quasi-Southerner such as Johnson could have been elected a priori, and Johnson had not-inconsiderable baggage as a wheeler-dealer/used car salesman anyhow. One suspects that a number of moderate northern Democrats, as much as they may have disliked Nixon, would distrust Johnson still more, and cast votes for the man they considered the lesser of two evils.
One interesting side note: without the Kennedys to throw the no-class rocks at Nixon, likely he is far less embittered and ostensibly paranoid. I don't see, as a result, anything remotely like Watergate in the 1961-69 Nixon presidency.
That also sets up 1968 as an interesting election: both presidential candidates might be from Massachusetts. Since Nixon was sort of groomed as an Eisenhower successor, so Nixon might have groomed Lodge.