On the Republican side, TR would have to come to some sort of accommodation with the more conservative wing of the party. Not the easiest task in the world, but brokered by such figures as Elihu Root and Henry Cabot Lodge, certainly feasible. Perhaps the deal would include having certain more prominent conservative types in the cabinet, assuming a GOP victory (e.g., former VP Charles Fairbanks as attorney general) or a not-quite-promise that Taft himself would be the next Supreme Court appointee.
On the Democrats' side, the convention would have to be all-but-impossibly deadlocked--and that's close to what happened at Baltimore's Fifth Regiment Armory in 1912 as it is. Wilson and Clark would have to hit high water marks just short of a majority, and neither would budge. Given Baltimore's enervating summer weather (I should know: I grew up there) and intransigence on the part of both major candidates, likely the delegates would be scrambling frantically for a solution of some sort. If the Democrats' convention took place after the Republican conclave in Chicago, a deadlock of this sort might well lead to an undercurrent of defeatism that would push the convention more quickly to a compromise candidate: in short, an unprecedented fourth nomination for Bryan. With Bryan's ego, he'd be largely blind to his less-than-optimum chances.
So we now have Roosevelt and generally acceptable running mate Missouri governor Herbert Hadley facing off against William Jennings Bryan and Thomas Marshall. The 1912 election will turn out to be one of the most one-sided in history, with the Democrats carrying only eight of the former Confederate states (TR's Rough Rider legacy would gain him Texas by a whisker, as well as narrowly carrying Virginia and Tennessee) plus the newer states of Oklahoma and Arizona. The rest: solidly Republican.
Woodrow Wilson, by the way, would finish his term as governor of New Jersey, but would not be re-elected. Out of work briefly, he would accept a post as president of the University of Virginia, returning to his native state. Wilson would be a constant critic of the Roosevelt administration, and would serve in the US Senate from Virginia, but would never again gather serious consideration for the presidency.