I think Eisenhower would play the role of kingmaker for the 1960 nomination honestly. As to who he'd crown, others on here would have a better idea than me. I don't see why the Democrats wouldn't still nominate Kennedy, though butterflies from a Nixon death might set the stage for someone else. As for the general election in 1960, it's a coin toss. On the one hand, the economy was kind of sour toward the end of Ike's term, so that helps the Democrats. On the other hand, despite that, and despite the flawed pledge to visit all 50 states and despite being off the campaign trail for some time due to illness or injury (I forget which it was of the two), Nixon still came damn close to beating Kennedy, so the Republicans still have a good shot (if not better odds than OTL) at winning in a Nixonless 1960.
As for 68, it depends on what happens in 1960 and 1964. If the Democrats still win in 1960, they likely hold on in 1964, so 68 would very much depend on who the loser of 64 was and how badly they lost. If it's like 1964 OTL where Goldwater loses and loses big, than 1968 would likely go to someone like Rockefeller or Romney. If it's a moderate/liberal that goes down in 64, a Conservative (maybe even Goldwater himself) gets the nomination.
As for American Conservatism itself, the Keynesian consensus that dominated since FDR unraveled in the mid to late 70s OTL, which was key to the rise/dominance of American Conservatism in the 1980s, 90s, and 2000s. My guess is you still have a rise of laissez faire / neoliberal economics sometime between 1976 and 2000, so in that sense Conservatism isn't any different than OTL. That said, it might not have been as dominated by cultural and social issues as it was thanks to Nixon and Reagan.