This is almost too devastating to contemplate but I'll try: for starters, this does nothing less than delay the emergence of the United States as a full world power for quite some time to come. Yes, the US stepped forward as a function of winning the Spanish-American War, but it took TR's mediation of the Russo-Japanese War and the Great White Fleet, as well as securing the independence of Panama to make the powers of Europe sit up and really take notice (apart from Great Britain, perhaps).
So: without TR, probably someone like Charles Fairbanks becomes McKinley's running mate in 1900 after Garrett Hobart's death. I doubt Bryan would have defeated McKinley, by the way. Assuming Leon Czolgosz was equally as unhinged in this timeline as in ours, McKinley is assassinated in September 1901, making Fairbanks president. Fairbanks was a reasonably competent sort, but nothing outstanding-probably capable but not a dynamic leader (in short, a less ponderous Taft).
Without TR, the anti-imperialists may get their way, and the US might not hang on to the Phillippines (ideally, Great Britain would step in, but that's open to question). A Fairbanks presidency would be pretty much like McKinley's. However, the progressive movement would have found itself without a true champion: Albert Beveridge of Indiana or Hiram Johnson of California could have stepped up but both were clearly of the second level as compared to TR.
In 1904, the election would have been a rather dull affair between two relatively conservative candidates (Fairbanks vs. Alton Parker); it's likely Fairbanks would have won a term in his own right, and likely would have worked to pave the way for Elihu Root as his successor (ironically, TR wanted Root to succeed him but Root declined citing ill health and his ties to Wall Street; I suspect the former was a red herring since Root lived well into his 90s, and the latter could have been overcome). Root would have been more progressive than Fairbanks but not nearly in the same league as TR.
More to follow.