I think if TR won in '12 then by '16 his "Progressives" would be running the Republican Party.
His victory could serve to move the whole spectrum of American politics in a more progressive direction, since Roosevelt men who have control of the GOP, the Democrats don't present a real alternative for "Taft" Republicans. The Taft men would have to stay in the GOP. Teddy's VP would have been Nicholas Murray Butler, the long-serving President of Columbia University.
Butler represented the progressive approach to government, with the focus on scientific solutions to the problems that government faced. I think the progressive control of the Republican Party is going to push men like Butler, Eilhu Root and Herbert Hoover into positions of power.
Roosevelt is going to push US involvement in WWI, though I don't know how successful he would be. If he is able to get the US into WWI, then he might run again in '16, and with the country at war he would win.
American participation (especially with TR as President) in WWI is definitely going to shorten the war. German sub warfare will be unlimited with American intervention, but with American industrial might behind the Entente I don't think the prospect of a staving England is ever going to be more than a pipe-dream for the Germans.
What is done with the American manpower? I think the "sideshow" campaigns in the Ottoman Empire would get a huge boost, either with American troops, or with more British troops since the Americans are on the Western Front. So the Ottoman Empire gets hammered in Palestine/Syria, Constantinople falls in '15, Russia stays in the game and the war is over by the end of '16?