I can't see Wilson sending Roosevelt to Europe in any circumstance. He's simply not going to allow one of his chief rivals a chance to further his celebrity. Now, assuming Roosevelt does not take his ill-fated trip to South America AND assuming his son is not killed, the Bull Moose will most likely be running for President in 1920. In 1920: The Year of Six Presidents there is litterally a chapter titled "He is the Only Candidate." That was a quote by a conservative Republican, whose name I cannot remember. The party was prepared to hand Roosevelt the nomination on a platter, and with it, the Presidency. His running-mate will either be Harding or Coolidge depending on how well Harry Daugherty (Harding's campaign manager) works the delegates. Regarldess, the Republican ticket will cruise right past Cox and the Democrats. Debbs might see his support up-tick with less Germans supporting Roosevelt, but that will have a neglible impact.
Teddy's job gets a lot harder once he's in office. Lodge and the conservatives aren't going to role over and let Roosevelt pass a second "Square Deal." He'll be forced to play hardball if he wants to get any major progressive legislation through Congres. The most likely outcome is that he sucessfully moves America from a wartime to peacetime economy, manages the recession of 1920-21 well, and gets credit for much of the early '20s prosperity. Roosevelt's foreign policy will be tougher towards Germany than Harding or Coolidge's was, possibly pushing that nation towards radicalism sooner. Regardless, Roosevelt will only help his legacy with another term in office.
The long-term ramifacations of a Roosevelt presidency in the 1920s, as is always the case in alternate history, are tougher to predict. It's unlikely that the Great Depression or Adolf Hitler are suddenly butterflied away. But you're going to have different Presidents in charge, and potentially significantly different polices in place. Assuming that Roosevelt does not run in 1924 due to his age and health, the Republicans will still be the overwhelming favorites to win. The battle for that party's nomination will be between some very strong candidates. Herbert Hoover may be the strongest, especially if Roosevelt throws his weight behing him. An earlier Hoover Presidency would be interesting to see. He was naturally a progressive, but of a very different mindset than FDR. His election may push the Republican's further to the left. Anyway, he'll win in 1924 and win reelection in 1928. If the Depression is not averted, than the Democrats will be favored to win in 1932. But an FDR presidency is not preordained. A lot of factors, not the least of which being his cousin's victory in 1920, could change Roosevelt's career trajectory. So, as I said at the beginning of this paragraph, we really can't predict anything after 1932.
Teddy's job gets a lot harder once he's in office. Lodge and the conservatives aren't going to role over and let Roosevelt pass a second "Square Deal." He'll be forced to play hardball if he wants to get any major progressive legislation through Congres. The most likely outcome is that he sucessfully moves America from a wartime to peacetime economy, manages the recession of 1920-21 well, and gets credit for much of the early '20s prosperity. Roosevelt's foreign policy will be tougher towards Germany than Harding or Coolidge's was, possibly pushing that nation towards radicalism sooner. Regardless, Roosevelt will only help his legacy with another term in office.
The long-term ramifacations of a Roosevelt presidency in the 1920s, as is always the case in alternate history, are tougher to predict. It's unlikely that the Great Depression or Adolf Hitler are suddenly butterflied away. But you're going to have different Presidents in charge, and potentially significantly different polices in place. Assuming that Roosevelt does not run in 1924 due to his age and health, the Republicans will still be the overwhelming favorites to win. The battle for that party's nomination will be between some very strong candidates. Herbert Hoover may be the strongest, especially if Roosevelt throws his weight behing him. An earlier Hoover Presidency would be interesting to see. He was naturally a progressive, but of a very different mindset than FDR. His election may push the Republican's further to the left. Anyway, he'll win in 1924 and win reelection in 1928. If the Depression is not averted, than the Democrats will be favored to win in 1932. But an FDR presidency is not preordained. A lot of factors, not the least of which being his cousin's victory in 1920, could change Roosevelt's career trajectory. So, as I said at the beginning of this paragraph, we really can't predict anything after 1932.