James Sherman would have assumed the presidency on Taft's death, but as noted, his health wasn't the best: he passed away himself in 1912. If we postulate Sherman dies also, then the presidency devolves upon Philander Knox, Secretary of State at the time (per the provisions of the act of 1886).
I somehow doubt Knox would have run in his own right in 1912; he was more the cabinet officer / administrative type. Also, I recall reading at some point that he was offered the 1920 GOP nomination but declined it (can't recall where, though). Those two suggest he'd have served out the term to which Taft was elected, and then bow out.
That clears the field for an all-out shootout between the progressives, mostly (but not entirely--see La Follette) united behind TR, and a fragmented conservative wing. Elihu Root turned down a chance to succeed TR in 1908 so I don't think he'd take the chance in 1912. Henry Cabot Lodge, while extremely effective representing Massachusetts in the Senate, probably couldn't win a nationwide election and probably wouldn't want to oppose directly an old friend in TR. Short version: it might take two ballots but TR gets the nomination, and picks a relatively conservative running mate (Charles Fairbanks redux, perhaps?).
Then it's a showdown between Wilson and TR...with the second instance of a presidential comeback after a hiatus ensuing.