An interesting notion. The 1896 elections are little noticed event in American History, sometimes. I find it odd they haven't garnered more attention this year in American media, but I digress.
The first problem with moving the Populist convention is that there's no guarantee that Bryan will be chosen. The OTL fusion occurred precisely because the Democrats' convention was divided: their first issue was to repudiate the pro-business, gold-standard policies of incumbent Grover Cleveland. After they did so, Bryan made his famous "Cross of Gold" speech. Only after this accomplishment did Bryan become the presumptive and eventual nominee--a profound feat for someone whose only political expereince in Washington was in the House and who hadn't served as governor of a large state. Bryan's nomination prompted the Gold Democrats to nominate their own ticket (John Palmer of Illinois and Simon Bolivar Buckner of Kentucky). The Populists nominated Bryan, but offered their own VP (Thomas Watson of Georgia). Without the OTL Democratic Convention, a dual candidacy is up-in-the-air. One could foresee somekind of circumstance which leads the Democratic convention to chose a compromise candidate to avoid a split in their ranks, Bryan being tainted by alleigiance of another Party.
For whatever reason, Bryan gets sick the night before the Democratic Convention and cannot deliver the Cross of Gold speech. The Democrats chose to nominate Arthur Sewall (who they nominated OTL as Bryan's running mate). Sewall is rather unlikely, having never held elected office, but re-nominating Cleveland is also unlikely. I'm looking for the Dems to nominate a relative no-body (probably a governor of a large state), though they will still run on a populist-inspired platform. Sewall is convienient because his nomination reeks of Party bosses connviving in secret and his personal wealth (he was a New England shipbuilder) was repugnant to the Populists.
Downcast, Bryan choses to attend the Populist convention. His voice recovered, he gives the most powerful speech of the day, the Cross of Gold speech. He becomes the Populist nominee. His fame and ties to the Democratic Party infuse Populist ranks. Thomas Watson of Georgia is the VP.
William McKinnley is nominated by the Republicans on a strict gold standard platform. Some delegates leave the convention to form the Silver (Republican) Party and they endorse the Populist ticket (similar to their endorsement of the Dem-Pop ticket OTL).
Determined to give the Populist party National prestige, Byran embarks on the first ever national speaking tour. He has to build a Party appartus from scratch as well. The Democrats lose support, given their accomodationist, compromise platform. Mark Hanna, however, trumps them all, running a traditinal front porch campaign with an unprecedent budget of $3M and hiring republican leaders to speak across the country (including Theodore Roosevelt).
In the general election, Populists carry Nevada, Colorado, Kansas, and Idaho. They also receive partial votes in California, Oregon and North Dakota. (Their OTL results in 1892). Due to the split tickets in the South, some southern states (Texas and Georgia) have very close elections. Much to the dismay of the Democratic establishment, it looks like the Republicans might break the Solid South; vote totals require a recount and chaos ensues, as in 1876. This prompts the Texas and Georgia legislatures to intervene, freeing their electors to "vote their conscience". (Alternatively they might simply order their electors to abstain). In reality, this means they will vote for whomever is not the Republican candidate. This leads to Bryan conducting a second speaking tour of the South, hoping that public rallies will convince delegates (i.e. Party bosses) to vote Populist.
Nonetheless, McKinley wins in the electoral college, with 271 votes. He takes only 46% of the popular vote (OTL it was 51%, but he's lost some with the Democrats compromising and the Silver Republicans). The Democrats take 32% and the Populists 20%. {The electoral count might be lower, but I'm trying to avoid a House Election, since it seems a bit too much given McKinley's strengths}.
The actions of Southern legislatures prompt outrage throughout the country, particularly in Populist camps and in the Progressive wing of the Republican Party. Bryan begins making a slow speaking circuit of the US, organizing for the Populist Party. He denounces the Electoral College and the indirect election of Senators. He elaborates on the imagery of the Cross of Gold speech, saying that the country has now been thrust into the Colliseum of the Elecoral College, where Senators, Party Bosses, and wealthy elites have conspired for their execution by Gladiatorial combat. The "Apostolic Speeches," so-called for the evocation of early Christian history, prove a powerful rallying cry. Nonetheless, the movement can do little but simmer...and organize.
McKinley's term proceeds on schedule as per OTL. The resurgent economy dampens Bryan's fervor, but resentment about "Judas Legislatures" still rankles. In mid-term elections, the Republican Party retains a majority of the House, though reduced, to 197 seats (-9). The Democrats have a major loss, from 124 seats to 28, with all 96 of those going to the Populists, who now have 122 seats (1 Seat is still held by a Silver Republican hold out and 1 by an independent, though both caucus and vote with the Populists.). In the Senate, The Populist delegation swells to 29, the Republicans with 44, the Democrats hold 17.
All of this means, that while the Spanish American war still breaks out, the anti-imperialist debates are much harsher. I'm thinking that the Populists probably are against the war and holding Cuba, Puerto Rico and the Phillippines, but it still happens.
From here on out, I'm not sure how the specifics turn out. It probably doesn't make sense to assume that McKinley is still shot, though TR is still probably VP in 1900. He and McKinley probably feud incessantly. Bryan runs again, though things stay as they were in 1898, since the Populists' objection to a successful war has them stymied. This makes 1904 a rather interesting contest. Let's say TR is contesting for the GOP nomination along with some establishment candidate (maybe Taft, maybe Charles Fairbanks, his OTL VP). This makes the GOP a contest between the Progressive Wing and the Old Guard. The Old Guard probably wins.
What does TR do? He may contemplate joining the Populists, by they have re-nominated Bryan for the third time. I think in 1904 he probably bolts and forms his own Party; this may be a stretch, since he could always bide his time and it leaves us with 4

parties! If he joins the Populists, it will be because the Populists have retained their tradition of a late convention and TR sense he can win something because the Populists are getting a bit tired of Bryan. Also, he agrees with the gist of much of the Populist platform (an activist government, direct election of senators, etc.). Even if it forms, I think the alliance would prove unweildy. If TR forms his own party, then we might see the Democrats and Republicans drift back together forming a party of conservative establishment, with two competing liberal/reforming parties.
It may work better if McKinley is shot (for whatever reason) and TR ends up in the White House. He will have more ample room for political maneuver and breathe life into the Progressive cause. If outcry is strong enough, the multi-party situation might become more permanent since the Electoral College might be seriously reformed given the "Judas Legislatures." I'd expect to see an expansion of the House as a compromise; perhaps MMP or Preference voting introduced in some way. Addmision of the western states will be a big issue, mostly for the senate since 6 extra seats for the Populists leaves no one with a majority.
All of this assumes, however, that the Democrats feel inclined to commit collective organizational suicide. Perhaps in 1904, they join the Populists, but keep the Populists' name. Can't you see all kinds of great jokes for later cynical Americans about the only difference between the Populists and the Republicans is one inch (i.e. the tail on the letter R)? This gives TR a bit more room to do his own Party-forming, but I'd imagine it might give his progressivism greater weight within the Republican Party as a counter-point to the Populists.