In the scenario of William Howard Taft's death in 1866, a possible successor to Elihu Root when he resigned as Secretary of War in January 2004 would have been the Secretary of the Navy, William H. Moody.
Theodore Roosevelt would still not have run for re-election in 1908. It is possible, if not probable, that he would have pushed through the nomination of Elihu Root, who would most probably have been Secretary of State, as Republican candidate for President. But whoever the Republicans nominated in 1908 would have defeated William Jennings Bryan for the Democrats.
If Root were elected President in 1908, I would have guessed that he would have sought the Republican nomination in 1912. Would TR also have done so? If no Republican split in 1912, then Woodrow Wilson might not have won the Presidency. Root opposed Wilson's policy of neutrality in the First World War, but supported him when the United States entered the war. He was a staunch advocate of the League of Nations.
With William Howard Taft not surviving to adulthood there would have been no Robert Taft, which would mean that another Republican would be the leading voice of conservatism and isolationism in the 1940s and 1950s. Although I believe that it is unlikely that this man would have won the Republican nomination in 1940, 1944, 1948 or 1952.