Taft will get a progressive primary challenger, but I doubt ther would be a viable third party run without TR's appeal. LaFollete did respectfully in 1924, but that was after Roosevelt's performance in 1912.
So with the national popular vote, IOTL:
Wilson 41.8%
Roosevelt 27.4%
Taft 23.2%
Debs 6.0%
Lets try these figures:
Wilson 45.8%
Taft 44.6%
Debs 8.0%
Is this plausible? Bryan got 43.8% in 1908. This represents a 4.5% swing from the Democrats to the Republicans. These totals still give the Democrats control of Congress. Also for comparison, Bryan got 46.7% in 1896.
Wilson could win both California and South Dakota, both of which went for Roosevelt IOTL. Taft doesn't seem to have been on the ballot in South Dakota, and got less than 4,000 votes in Caliifornia. California in particular was close. The two combined for 18 electoral votes.
Taft still wins the other four Roosevelt states (Pennsylvania, Michigan, Minnesota, and Washington), plus his two states of Utah and Vermont. Pennsylvania, Minnesota, and Vermont did not vote Democrat between the Civil War and Great Depression, and Michigan voted Democrat only once in that period.
Taft will pick up the other five New England states, for 40 electoral votes. Wilson took well under 40% in all of them. Wilson probably loses all the states where he took under 40% to Taft, except for Nevada which had a large Debs vote. The other states and Kansas, Wyoming, Illinois, Montana, Oregon, North Dakota, Iowa, and Idaho for a total of 73 electoral votes. Ohio, with 24 electoral votes, is really close. IOTL Wilson got 41% of the vote, to 49% combined for Roosevelt and Taft, with Taft doing better than Roosevelt.
So take the combined Taft and Roosevelt electoral votes of 96, subtract California's 13, then add 113 for the states, except Nevada, where Wilson polled under 40%, and add 24 for Ohio. This is 220 electoral votes for Taft as opposed to 331 for Wilson. It looks like Wilson wins, carrying all the southern and border states, New York, New Jersey, and Indiana. Nebraska, and several western states, possibly including California. Wisconsin either votes for Wilson or a LaFollette favorite on candidacy.
However, New York, New Mexico, and New Jersey are extremely close. The combined Roosevelt and Taft margin over Wilson was 10% in New Mexico, 11% in New York, and 12% in New Jersey. Taft could just about swing it if he draws a straight of New York, Ohio, and either New Jersey or California. The problem is that I just don't see him winning New Jersey or California against Wilson, though Hughes did carry New Jersey against Wilson in 1916.
Keep in mind that in this scenario, Roosevelt will likely be campaigning for Taft, since it would strengthen his position in the party in 1916. And that will help.
I did a similar analysis of Roosevelt's chances as the sole Republican candidate in the other thread, and I like them better. I concluded the election would come down to New York in that thread. Taft also needs New York to win, but he needs lots of help elsewhere. I think the additional additional 4% cushion in the national popular vote Roosevelt starts with would make up for Taft votes defecting to Wilson in this scenario, though it would be close.