What happens to FDR and his New Deal in a TL where TR wins the election of 1912 as a Progressive? If there is a minimal amount of butterflies involved and the world still sees a Great Depression around the 1930's, does FDR still have a chance to become President someday, and will America still see something like the New Deal?
I think, that in earnest, a third term for Theodore Roosevelt means that the 'New Nationalism' does a lot of what the New Deal does, but much earlier. Unions are recognized, government does some economic planning and whatnot. Then again, there's probably a lot that isn't addressed. Barring an economic panic, there's probably no Wall Street regulatory apparatus put into place, which likely means you'll still have a Depression anyway, though because of a social safety net being in place, it might not be as hard on the common man or woman.
As for Franklin Roosevelt, I'm not exactly sure where he ends up. I would say that Teddy (his personal hero, no doubt) probably gives Franklin a job in the administration even though F.D.R. supported Wilson. Assistant Secretary to the Navy, as per OTL, is probably where F.D.R. ends up. I figure that T.R. would enter WWI on the side of the allies at the outset of war, which could mean Franklin resigns his position to join up (OTL, the only thing that did was Wilson's refusal to let Franklin go to war). I doubt T.R. is going to block F.D.R.'s decision, and supposing that F.D.R. survives, he comes back with a lot more experience in military matters than he might have otherwise had in OTL.
If Roosevelt dies in 1919 as IOTL, Johnson takes over as President (of course, this is assuming a fourth term for T.R., which I don't see as too much of a long shot) with the war drawing down. By this point, the Progressives have replaced the Republicans as the second party of the republic, leading big business elements into the Democratic Party. Isolationists probably beat Johnson in 1920, leading to a Democratic twenties and likely, the same sort of collapse we saw in 1929, only under someone like Al Smith, rather than Herbie Hoover.
My biggest question, though, in all of this, is whether or not F.D.R. remains a Democrat or becomes a Progressive. IOTL, Roosevelt became a Democrat largely because of his father, who was close friends with Grover Cleveland. Of course, with his cousin leading the Progressive movement into a separate party, the erstwhile Progressive F.D.R. might feel uncomfortable in a party dominated by business interests. I can see Roosevelt possibly crossing over into the Progressive Party, in that case. If that happens, I'd say he'd eventually be a shoo-in for their Presidential nomination in the same way he was for the Democrats: [a] He's a Roosevelt.
He's a charismatic leader and a great speaker. So, yes, to answer your question, I think it's likely for Roosevelt as for any other man to be the Progressive leader of the ticket in '32.