WI an amendment in the terms of th e 22nd had been passed earlier, perhaps out of fear of TR.
The first big effect of this I think would be paradoxical. In OTL Cooleridge did not run because of the 2 term tradition. However the 22nd would actually have allowed him to run in 1928 (he got less than 2 years of Harding's term.
I wonder how different FDR would have behaved if he were not allowed to run in 1940.
One point. Something like this very nearly did happen. See
http://query.nytimes.com/mem/archive-free/pdf?res=FA0F15FC385F13738DDDAB0894DA405B838DF1D3
This article, limiting the POTUS to a single six-year term, passed the Senate in Feb 1913, but never went to the House because President-Elect Wilson decided that he did not approve of it (despite it's being part of the Democratic Platform on which he had been elected), and wrote to the relevant Committee Chairman to get it buried.
Had Wilson OK'd it [1] and the House passed it unaltered, then the next Presidential election would have been in 1918, with subsequent ones in 1924, 1930, 1936 - -
However, there were suggestions that the Amendment should not apply to the POTUS holding office when it came into force, and the HoR might so alter it. In that case, it would be effective from 1916, and thereafter 1922, 1928, 1934 etc. That version is a bit worrying, as it probably keeps Hoover in office two years longer, during which time things could get nasty. However, FDR is very likely elected in 1934 (unless the Dems for some reason nominate someone else) and we get the New Deal much as remembered.
As for 1940, it's really anybody's guess. Henry Wallace is one obvious possibility. Fotrunately, this Amendment means that he leaves the White House for good in 1947, hopefully before his softness on the SU has time to do too much harm.
[1] Or had someone else been President. The likeliest alternatives, Champ Clark and WJ Bryan, both supported it. I'm not sure if Thomas R Marshall was on record, but he was a pretty partisan Democrat ansd would likely have gone along with the Party latform.