The conservatives regained the upper hand in the Dems by 1904, meaning Bryan-or anyone allied closely-won't get a shot. It would mean either Alton Parker, as IOTL, or perhaps former SecState Richard Olney. William Randolph Hearst (yes, same as the newspapers) was a NY congressman who tried for the nomination but didn't make it. From all I can see, the Dems' field of candidates in 1904 was pretty weak, meaning it would have been all but a walkover for TR no matter what.
By 1908, though, the Bryan wing of the party was back in power. That would probably mean Bryan gets the nod again (IOTL, the other two candidates were pretty insignificant: former DE senator George Gray and MN governor John Johnson). I'll maintain TR would have turned Bryan every which way but loose, with entirely G-rated (by modern standards) but very creative English, and would have won a second term rather handily.
By 1912, not only is Bryan a three-time loser, but I could see him as something of a joke in his own party, much the way that Harold Stassen became IOTL. He won't have that much influence at the 1912 convention, so it'll come down to a dogfight between Wilson and Champ Clark. Without Bryan's opposition to Clark--a result of a Tammany Hall endorsement IOTL--Wilson has a tougher time. Even with Bryan's help, it took 46 ballots to give Wilson the nod; under these circumstances, I could see a deadlocked convention going for a compromise candidate, like Ohio's Judson Harmon (yes, I reconsidered my position earlier). Opposing Harmon will be Elihu Root, who could legitimately be called a Wall Street candidate. But with TR's backing, I think he could have pulled it off...narrowly. There was just enough of the progressive in Root to want him to have TR handy, and to continue a lot of TR's policies, albeit somewhat soft-pedaled.