To get to this scenario, I have always had Taft get a health problem. Considering his weight, that could easily happen. TR gets the nomination and wins the election. His desire to go to war gets him in trouble. If he runs in 1916, a Democrat campaigning on the I will keep us out of war platform defeats him. The unrestricted submarine warfare and the Zimmerman telegram force that Democrat to ask for a declaration of war. In 1920, the angry electorate defeats the President in a landslide. Warren Harding is still the best compromise candidate for the 1920, Republican nomination, Calvin Coodidge is still the idea running mate for Harding.
So
1909 - 1913 William Howard Taft
1913 - 1917 Theodore Roosevelt
1917 - 1921 Some Democrat
1921 - Same as OTL