I think he runs in the 20s, gets spanked by Coolidge, ends up retiring to his estate and in 33 some other dem, beats hoover. The butterflies here are amazing however, what if we get a fiscal conservative dem rather than a progressive dem? The Hoover policies are repealed and not exploded. The depression ends in 35, dems sweep in 36, still take a hit in 37, but not as bad for no court packing, but by 1940 a republican takes over with slim majorities in the house. War breaks out. But what kind of republcan won? a TR type, or a Cooldige type?
What did the non FDR dem do about the empire of japan's agreesion in China. Did Japan still feel the need to stop the USA? Probably.
Also, remember Eleanor Roosevelt did a lot for social stuff. She reached out to the African Americans who in 36 made a major leap to the Dems. with out her, or the job handouts of the depression, African Americans will still be looking at both parties, republicans as the guys who abandoned them, and dems as the guys who ignore or lynch them. This has major butterflies for the 1960s because It is very very likely that the USA gets involved in WW2 and very likely comes out on top. Meaning There will still be thousands of black soldiers and they come back looking to cash in. the Double V still happens (victory vs fascism and victory vs segregation) Since Truman and Ike both made major strides towards civil rights it is very likley that the post ww2 presidents (Ike may still be there, unless he runs afoul of the president and doesn't become supreme commander) that Civil Rights will still be pushed because of the optics of the cold war and tens of thousands of idealistic post war baby boomers in college reading about how "All men are created equal" and then wondering about why they had to stand in a different line.
Without FDR's new deal, (Assuming we don't have some other new deal) the Unions wouldn't be quite as strong because the Wagner Act was one of the key issues. This would alter political dynamics all across the next century on everything from trade issues, ecological laws, political contributions, and education.
We wouldn't have the 22nd Amendment in the 1950s and probably not until some other president broke the unspoken rule about 2 terms.
Who ever is elected in 1940 has to deal with Stalin in such a way to prevent the cold war from becoming ww3. Perhaps he will be able to contain stalin some way and change the cold war all together. From what i recall FDRs death and Truman being kept in the dark, along with Churchill being fired allowed Stalin to be the big man of the allies and get more of his way.
Of course, in 1940 the USA might elect an isolationist Jeanette Rankin type who might only go to war with Japan after Pearl Harbor. Japan will be defeated, probably not nuked. Europe Falls to the USSR, and Soviet France is established. Maybe even Soviet Spain. The Cold War is radically different at that point. Hell, the USA might even start falling to internal communist elections and rioting with the upheavals of the 1960s.