Interesting and not implausible premise. Let's see...by 1956, although the Cold War was an everyday presence, it wasn't as glacial as it had been in 1952 (OK, the Bulletin of the Atomic Scientists, as it was called then, still had the Doomsday Clock at 11:58 but things weren't as grim as they had been); McCarthy was in eclipse and would be dead within approximately a year; the economy was doing pretty well. On the surface, Ike could pretty much name a successor although that wouldn't be his style.
Would Nixon be the nominee? My gut instinct is no. Ike would probably throw his influence (quietly, of course) behind someone else: guessing a third time around for Dewey. This time, with the entire party lined up behind him and Ike's prestige backing him, I think Dewey carries it off. Perhaps Nixon would remain as VP with the tacit understanding that it would be a steppingstone for '64.
A Dewey administration from 1957 onward would presumably be similar to a second Ike term, albeit perhaps with a bit more emphasis on the northeastern establishment / Wall Street. Expect Foster Dulles to continue as SecState; perhaps Henry Cabot Lodge Jr would join the cabinet (maybe as Brownell's successor at Justice?). In any event, with Dewey in the Oval Office, perhaps civil rights might have been accelerated modestly with respect to OTL: I could see him sending troops to keep order in Little Rock, and taking a somewhat more hands-on approach.
Dewey might have some difficulty in re-election in '60, assuming there's still a recession in '58. But even if he faces John Kennedy, and even with his near-zero icy personality, he still has an incumbent's advantage. I'd look for another squeaker in '60, but this time with Kennedy and all the Cook County (IL) tombstones on the (just barely) losing side.
The Dewey presidency would, IMO, forestall a shift toward the right of the GOP, and at least prolong the control of the party by the northeastern moderates. Nixon in '64 would be acceptable/tolerable to both factions, particularly if teamed up with someone like Bill Scranton.
That could prove a fascinating, if particularly dirty, election: Nixon vs. Lyndon Johnson, since Kennedy might well still be licking his wounds from '60. Either way, the loser would have trouble keeping a straight face in the concession speech, all the while itching to call his opponent something like "...that son of a bitch..." for openers. You can bet that would have as much mud-slinging as 1928 as a minimum.