You have a GOP whose candidates are so deeply unpopular they had to recruit from outside of the political sphere. Assume Ike doesn't run, Stevenson has a real shot.
The other timeline about this is quite good, with good analysis before the butterflying gets big and some premium grade Crazy when the butterflies really start going. A+ stuff.
Politically though, I wonder if you get a Goldwater type of deal starting earlier, perhaps pacing Brown and the first wave of the Civil Rights movement. The reasoning behind this is that you get enough solid desperation in a Republican Party that keeps getting beat and soon enough the kooks are going to seem appealing. The GOP had seen success with vicious red-baiting in the late forties and 1950. If the adults in the party haven't won a Presidential since 1928, the hard right is going to seem like the only part of the party that's had any success.
On the other hand, continuing Democratic dominance, combined with a more radical right, might make the New Deal even more of the bed-rock of American life. A more firmly ensconced, but perhaps less innovative, Democratic Party arrives in 1960 and who knows?