Well if TR is President in 1908 you probably do away with the tariff issue that caused such a schism in the Republican party at the time. TR had stopped publicly talking about it at this time because of the reactions and possible party split. It also means American Steel, since its creation TR approved is not busted up.
This means that the Republican party stays more moderate if TR is President. He was not the bullish conservative that Taft was. It is possible TR uses a Supreme Court appointment to Chief Justice to mollify the conservatives while also silencing them in effect by getting Taft out of politics and into the court.
As for who would run in 1912, Charles W. Fairbanks. Without the split in the Republican Party he'd be favored over Charles Evan Hughes. Coming right off of TR's presidency he would probably easily win a first term as I believe TR would maintain most of his popularity and Fairbanks was his Vice President. Without the split in the R I believe he'd beat Wilson easily.
Now the question is how does WWI now differ with Fairbanks in charge. During a war generally the incumbent President always wins so Fairbanks wins again in 1916 (However his health will kill him halfway through his term leaving his Vice President as President for the next election).
This is my guess James S. Sherman was the Vice Presidential nominee on the 1912 ticket. His death in October of 1912 opens a quandary. Its just before the election but not enough time to reprint ballots. He's dead so the Senate is unlikely to recognize him as a valid candidate when the electoral votes are read. That means the only valid VP candidate with electoral votes is Thomas R. Marshall. The problem is that both Fairbanks and Marshall are from Indiana, so there is no valid VP candidate, so I believe its left vacant.
In 1916 Fairbanks vice president is probably Elihu Root. Whether they can defeat Wilson now I'm not sure but generally the War President wins if he's the incumbent so my bet is that Fairbanks/Hughes win narrowly. in 1916.
You've now had a R in the Presidency for a very long time in 1920 the Democrats will be poised to win the Presidency.
James M. Cox is probably still the most apt to be nominated and he'll win the country gets tired of having one party in the WH over time. Franklin Delano Rooselvelt is his vice president. I believe they would easily be reelected. Also Cox/Rooselvelt could run for a 3rd term and citing TR's precedence probably does. You've avoided the bubble of the 1920s that collapsed since Coolidge is not in the Treasury and not president during this time.
FDR runs in 1932. Garner is VP as it was a deal needed to be made, and the other major contender is the same state as FDR.
FDR has now been in Washington 16 years, his health is failing and he is getting to be frail. He decides not to run again in 1940 as a result, even if he did he'd never live out to see the end of the term. There would be a bitter battle between James A Farley and James N. Garner over the nomination, this opens them up to sadly a Republican takeover of the White House.
Wendell Willkie secures the Republican nomination after an outpouring of letters in support of aid to Britain he wins the election. Charles L. McNary is his Vice President. C
1909-1913 Theodore Roosevelt/Charles W. Fairbanks
1913-1917 Charles Fairbanks/Vacant
1917-1921 Charles W. Fairbanks/Elihu Root
1921-1933 James M. Cox/Franklin D. Roosevelt
1934-1941 Franklin D. Roosevelt/James N Garner
1941- April 1944 Wendell Willkie/Charles L. McNary
April 1944-Oct 1944 Wendell Willkie/Vacant.
Oct 1944 We have a problem Houston......the President has died with no Vice President.....as he died 6 months before.
Oct 1944-1945 Thomas E. Dewey (former Sec of State under Willkie)/Vacant
1945-1953 Harry F. Byrd/Harry S. Truman
1953-1961 Dwight D. Eisenhower/Richard M Nixon
This is going to be hard times for the R, since both the Vice President and President died in 1944. Wendell Wilkie would have been the nominee for President on the ticket, with McNary's death earlier this would make John W. Bricker the most likely candidate for Vice President with the death of Wilkie he'd be the Presidential Candidate now with no Republican Vice President on the ticket. (its really to late to reprint tickets...)
The most likely nominee in 44 for the democrats is Harry F. Byrd with Harry S. Truman as his running mate. I believe Byrd/Truman would win mostly due to the chaos caused by the death of Willkie. They push through an Amendment to make sure a line of sucession exists