A heathier T.R. could have run and won in 1920. The Brazil trip in 1914 broke his health, and contributed to his early at age 60 in janaury of 1919.
Given the lack of a clear Republcian front-runner after his death, it is likely that Roosevet would have been nominated by the Republican Party in 1920. Given the actual Republcian landslide under the previously little-known Warren G. Harding in 1920, one could expect Roosevelt to win handily that year as well.
As for running mates.Franklin Roosevelt doesn't make it onto the Democratic ticket led by Govrenor James Cox. T.R. is probably paired with a midwesterner, such as Warren Harding of Ohio.
Another T.R. presidency starting in 1921 would be very different than that of 1901-09. Ironically, the United States might have joined the League of Nations under President T.R., albeit with the reservations sought by T.R.'s old friend Senator Lodge. A conservatove pst-war Congress would have stymied further progressive reform.