Wide Open in 1960
What makes Nixon not want to run? He really wanted it, and he was favored to win it all the way until the famous debate with Kennedy.
The only way Nixon doesn't run in 1960 is: (a) he dies; or (b) he suffers a debilitating injury or health problem. Perhaps, for purposes of this alt-history timeline, we can say that Nixon is killed in Caracas, Venezuela when a communist-inspired mob attacks his motorcade there in May of 1958?
The November 1958 mid-terms were a Democratic "wave election" and the Republicans went from near-parity with the Democrats in both Houses of Congress and in the number of state governorships to being the clear minority party after that election. Wins by Nelson Rockefeller for the governorship of New York, and by Senator Barry Goldwater for re-election in Arizona, were among the few bright spots for Republicans in that election year.
Nelson Rockfeller would have run for president in 1960, as he had originally intended. It was Nixon's strength within the party which led Rockfeller to abandon his plans to run for president at the begining of 1960 in OTL anyway. Senator Bill Knowland's landslide defeat for governor of California in 1958 ruined his own plans for 1960. Goldwater probably would not have run in 1960, as he was a reluctant presidential candidate even in 1963-64.
Although the Republican "bench" was decimated by 1958 mid-term election losses, I doubt Rockefeller would have run uncontested for the 1960 Republican nomination as Nixon did in OTL. Perhaps, with Nixon out of the nomination picture, Ike persuades his own favorite, Secretary of the Treasury Robert Anderson, to switch parties (Anderson was a conservative Texas Democrat) and run as a Republican with Ike's overt support. Perhaps Senator Thurston Morton of Kentucky and other Republicans run.
Still, it is difficult to see who can stop NY Governor Rockefeller and his big-money campaign machine in the primaries. (1960 was before Rockfeller's then-controversial divorce in 1962 and re-marriage in 1963, and before his big-spending and big-taxing policies as NY governor became established).
Rockefeller vs. Kennedy in 1960 would have been a very interesting race. One obvious Democrat strategy of attacking Rockefeller's wealth would have been severely hamperred, or scrapped entirely, given the Kennedy family's own wealth. Rockefeller may have carried NY State (with its then 45 electoral votes), but Kennedy would have run stronger in California (then 32 electoral votes) without Nixon on the Republican ticket.