Although it's possible for him to win in 1920, it's unlikely. The country grew tired of the progressive reforms and foreign interventions. They wanted to return home and focus on business.
Even if he wins in 1924, I also doubt the Democrats can win in 1928. The country was going though its greatest boom period ever (unknown how TR would affect this) and the Democrats were deeply divided. They nomainted Al Smith, a catholic. This led to lots of southern and KKK members to walk out the convention. And he loss to Hoover, one of America's greatest heroes at THAT time. He had a rags to riches orphan story and saved thousands of lives after WW1.
EVEN if the Democrats win in 1928, I SERIOUSLY doubt the GOP will nomanite TR in 1932. Lots of people were against a 3rd term and although he might get away with running in 1920/1925, people would call foul if he ran again. I doubt he'll win. Hell, people might blame him. No way he keeps winning past 1936 even if he gets credit for the recovery.
Of course, having a progressive president might butterfly away the Great Depression or make it more mild. But then you get rid of the roaring 20s. This dramatically changes WW2, if there is one. That's the only reason FDR ran in 1940/1944 OTL