If Teddy gets the Republican nomination, he wins the Presidency.
But from there, things get interesting. Without Roosevelt running third party, Wilson (or Champ Clark, who is probably more likely to get the Democratic nomination ITTL) gets smacked down, and the Democrats now haven't held the White House in twenty years. That means a circular firing squad within the Democratic Party. If Champ gets the nomination in '12, Wilson gets it in '16 by going after the mainstream party organization for not being 'progressive' enough.
I don't think Roosevelt gets the United States into the war in 1914. Public support just *isn't* there. He might be able to swing it in 1915, though. If the United States enters in 1915, Teddy beats Wilson (who isn't exactly running as an opponent of the war ITTL) and a weaker left-wing challenge from the Socialists, who are really still the only party opposed to waging the war.
By 1918, the war is over or ending, and the Democrats take back control of Congress. The Socialists likely face the same wartime repression they faced under Wilson (don't think for a minute that Roosevelt wouldn't have done the same) and are effectively defeated, for the time being. Roosevelt wins the Republican nomination again in 1920, ditching his running mate Hiram Johnson for Senator Warren Harding of Ohio.
The Democrats, who renominate Wilson, run against Roosevelt's war policies and basically project a 'Had Enough?' style campaign against the Republicans. Wilson has a pretty easy time portraying Roosevelt as a would-be-king in seeking his fifth term in office, and defeats him pretty spectacularly. The Democrats return to power for the first time since 1892.
Wilson thus presides over a depression in the early 1920s, a wave of strikes, and a lot of unpleasantness. He dies from a stroke in 1923, allowing Vice President James Cox to take over. Cox takes office as the economy is rebounding, and faces off against Republican Harding and Socialist Eugene Debs in November, winning a pretty strong mandate for another term.
Cox declines to seek a second term in 1928 and is succeeded by Franklin Roosevelt, his Vice President and the former Senator from New York. Roosevelt defeats Republican Calvin Coolidge pretty handily. Of course, a stock market crash mires the Roosevelt administration in a Depression that leads to Roosevelt losing the White House in 1932 to a certain Herbert Hoover...