Assuming Hobart had lived, and had been re-nominated, he would have succeeded McKinley in 1901, all else being equal. But all that would do is delay a TR presidency to the 1904 or at worst to the 1908 elections. In the meantime, it's reasonable to assume that TR would have some sort of cabinet post (Secretary of the Navy or even Secretary of War) in a Hobart administration.
Actually, it's not a guarantee that TR has a Cabinet Post. If he'd lived, Hobart's re-nomination is pretty much a lock, since he and McKinley got on so well; Hobart was actually so influential in McKinley's administration that he was called "the Assistant President."
Whether McKinley is shot is probably influenced by one's take on the butterfly theory, but I can't see a good reason why he wouldn't go to the Pan-American Exposition at Buffalo nor why Leon Czolgosz wouldn't still be inspired by the assassination of Umberto I of Italy in July 1900. Hence, for the sake of argument McKinley is still shot and still dies.
Thence, Garret Hobart becomes President. Rather than a young, Progressive, out-going trust-buster, you know have someone with William McKinley's and Mark Hanna's views on politics, but who'd never felt them strongly enough to pursue high office. The VPcy was almost foisted on him by the New Jersey Republican delegation. He is the ultimate version of "His Accidency."
TR probably runs for another term as Governor of NY in 1900, which further puts the Old Guard Republicans there on edge. His term would be up in 1902. He might try to challenge Thomas Platt for Platt's Senate seat, but it would seem to me that Platt could prevent that by getting TR appointed Secretary of the Navy. OTL John Long resigned in 1902. He might not TTL, since his resignation might have been prompted by TR's ascension, but he'd been a do-nothing secretary to begin with and Hobart had managed McKinley's Cabinet fairly well. Replacing Long with TR is great for expansionist Republicans and keeps TR away from domestic politics.
Now this is all further complicated by the Coal Strike of 1902 (which started in autumn or so). This might be enough to let TR challenge Platt, but he might not be able to do so, given Platt's control of the NY Legislature. It probably isn't enough, so we'll leave the above arrangement. However, without TR the strike will have to be settled in a different way. In 1898 and 1900, Mark Hanna had actually played a key role in getting management to make concessions to the unions because of the coming war and election. JP Morgan also stepped in 1902, so I'd imagine that in the absence of TR, Hanna and Morgan manage the situation. However, it won't result in the Anthracite Coal Commision nor will it mark a new acitivism for the executive branch in regulation. Miners are no doubt not as appeased and their demands might linger; Socialists might gain a bigger foothold--particularly since the sitting President used to be a corporate lawyer.
As Secretary of the Navy, TR still gets to have a hand in the construction of the Panama Canal. The Hay-Pauncenforte Treaty is probably signed on schedule in 1901. However, then there's the great question of whether the Nicaragua route might win out. Certainly, this is open to butterflies. It would seem to me that French syndicate would still attempt to lobby Congress to chose the Panama route (so that the US would buy them out). However, if TR is in charge or has more time to devote to the project, perhaps he can take a closer interest and decide to take a tour to pick the area himself. He might chose Nicaragua since it wouldn't require engineering the independence of Panama, which would be harder for TR to manage as only SecNav (though Hobart may well be for it as well).
Now we get to the election of 1904. Now, one issue to consider is whether the presidency has drained Hobart's health and / or whether he even wants a term in his own right. If Mark Hanna dies in February 1904, however, the Republicans may have few options. By 1904, TR may not be as popular as he was in 1900 and Thomas Platt has no reason to "kick him upstairs" since he's quite distracted at the Navy Department. Hence, I doubt he's the VP nominee, though he may be proposed by some. Let's go with Charles Fairbanks as the GOP VP. Alternatively, John Hay might be nice. The Progressive Republicans are rambunctious, but they haven't gotten the boost of TR's Presidency.
Now, the Democratic Convention will be interesting. OTL TR was hugely popular and the Democrats nominated Alton Parker as a sacrificial lamb, even though William Randolph Hearst wanted the nod. The Democrats should be hankering to win in 1904, particularly since they can paint Hobart as a business crony. No VP to ascend the Presidency had ever won a term in his own right. Hence, William Jennigs Bryan probably wants another go (his third); William Randolph Hearst may contest, but he was a supporter of Bryan's so I'd think he backs him. If Bryan can win the nomination against whomever the conservative Democrats chose, I think he's got a good chance to win in 1904. Now, Bryan might be forced to make some kind of a deal with the Bourbon Democrats in order to get the nod; most likely a conservative VP in this case (Alton Parker himself, perhaps).
The results are 1) liable to be close and 2) will probably have hinged on the handling of the 1902 Strike. I'm tempted to think that it's resolved with more bloodshed and hardship than OTL; at the very least, TR's Square Deal has not presented Progressive and reformers and those feeling oppressed with an out. I'm tempted to say Bryan wins, but he'd have to score huge wins in at least three of the following: PA, OH, IN, IL, WV. WV is easy, but OH and PA are solid Republican territory. For Bryan to win, it would have to be a watershed event for the Democrats, with lots of Progressive cross-over votes.
If Hobart wins, his health must begin to become suspect. TR will also need to find a way to gain national prominence and the Progressives will be hungry for change. If Hobart loses, TR has a chance to do some organizing and take the GOP nomination in 1908. Let's say a third-party bid in union heavy states leads to the narrow victory of Bryan.
As President, Bryan is hamstrung by conservative Democrats, but he might do a lot. If the canal project is not underway, he might cancel it; certainly, military spending goes down. If he can find a reason, he might start a time-table for Phillippine independence. Domestically, he could wreak all sorts of havoc by pushing regulation. He probably does some of what TR proposed, but he's more populist than progressive. You'd see a strong Dep't of Agriculture, but I can't really see any change on the Gold Standard. Probably a lot of talk without getting much done.
In any case, all of this is to say, I'd think the most natural time for a TR presidency in this case in 1908. He serves through 1917 at least and if a WWI situation breaks out, he could easily campaign for a third term, though he might die in the midst of it.
