I'm inclined to believe that even without his election night pronouncement in 1904, he would have stepped aside, however briefly, in 1908, out of respect for the tradition even if he didn't have two elected terms under his belt by then. In effect, he had, since he served all but six months of the term from March 1901-March 1905, but I said that to keep things straight.
Anyhow: if he had persuaded Taft to accept a Supreme Court appointment, my sense is that Elihu Root would have gotten the 1908 nod. True, as a consummate Wall Street lawyer, Root would have been a tougher sell than Taft, but with Bryan peddling his shopworn, tired message for a third time in 1908, and having TR stumping on his behalf, Root would have won.
I don't think Root would have had the same controversies that Taft encountered, leading to a falling out. On the other hand, since, in OTL, Root begged off entirely on the grounds of health, I could see Root quietly and behind the scenes stipulating he'd serve one and only one term. That sets the stage neatly for a TR return in 1912.
Come 1912, a united GOP (say, TR and MO governor Herbert Hadley) take care of Wilson/Marshall fairly neatly. TR goes on to serve two full terms (no Amazon basin trip in 1916 means no debilitating illnesses that led indirectly to his death in 1919) and retires in March 1921, quite possibly in favor of Charles Evans Hughes, who would have been his Secretary of State (and that presumes that Hughes' daughter would not have passed away from tuberculosis: perhaps better medical care or some similar departure).
Yes, I think 1920 would still have been a GOP year: you don't go against a singularly popular president who won his second Nobel peace prize in 1915 for getting Austria-Hungary and Serbia along with the major powers of Europe to the conference table to settle the Austro-Serbian crisis of 1914 without it going hot. And TR had enough prestige in the chancelleries of Europe to pull that off, whereas Wilson was essentially unknown to them.