Teddy living past 1919 ensures that he'll be President again come March 4, 1921. There's no going around that. He had made his peace with the Republican establishment by that time and was the clear front-runner going in. Without progressives divided between Wood and Johnson and La Follette (but then again, the La Follette types were never going to rally behind T.R.), Roosevelt could win the nomination over the conservative Old Guard.
This of course probably butterflies FDR out of the White House. TR and FDR had a very cordial relationship during the Wilson years, as FDR was feeding internal info to TR while he served as Assistant Naval Secretary. If anything, TR at the top of the Republican ticket means the Democrats won't nominate FDR for Veep (nor would he accept the spot with his cousin heading the GOP ticket) and instead choose someone far more lackluster. TR will probably appoint FDR to Secretary of the Navy in his own administration, and Roosevelt serving there (assuming TR lives through his first term and predictably seeks another in '24) there won't be anyone to dash Junior's hopes of the New York Governorship in '24, either. Junior might still lose to Al Smith, though. If Smith wins, you can bet Junior will probably go about his life the way he did OTL. FDR probably resigns as Naval Secretary at some point if Junior doesn't get elected Gov and runs himself, probably in '28.
TR's third and fourth terms probably see some legislation that brunts some of the economic pain of the forthcoming depression, which was pretty much inevitable after the farm crash of the 20s. With the farm sector doing better, it might very well turn out to be a shorter affair, especially if TR enacts much of his 1912 platform in the 1920s. TR's vice president, Warren Harding, is elected in '28 over Al Smith, though he loses in his bid for a second term to the former Naval Secretary and Governor of New York, FDR.
With less to do thanks to his cousin, Roosevelt probably sticks to economic stimulus and actually probably is more successful pushing a planning apparatus in the U.S. economy like the NRA. Either way, a shorter depression or a less bad one will weaken some of the impetus for further reform, especially if some of it gets enacted under Teddy. The end result might be a more corporatist FDR administration that is in some respects more conservative, only because much of the heavy-lifting has been done by his cousin in the 1920s. TR would probably be pretty happy with this, and assuming World War II breaks out on schedule (and TR is still alive) he'd be all in for his cousin running for a third term.
Party politics would certainly be interesting in such a TL.
