Hmm, methinks this inspired by the thread on the Populists now in the Pre-1900 forum.
An interesting question in any case. First, there's the potential for the 1896 election to create a more genuine Populist party. The easiest way is for the Dems not to nominate Bryan. I'm not sure if the Populists immediately would, since Bryan was a loyal Democrat (as others have pointed out) and since he wasn't quite so famous before the Cross of Gold speech and the 1896 campaign. The Democrats run a compromise candidate instead, to avoid a split on the Gold Standard question, but this only hurts them as they lose out to Populists.
William McKinley meanwhile takes power as an Old Guard GOP president. There's no guarantee that TR rises to office as we know him to have done: certainly a TTL assissination/sucession is a bit deterministic, though not out of the question. If the Populists survive, however, TR's leadership of the Progressive wing of the Republican Party has interesting effects. First, Progressivism might end up being a useful counter-point to Populism (the difference being urban v. agrarian). If three party politics continues into the early 20th Century, than any number of re-orientiations is possible. Perhaps TR leads progressives into the Democrats to take advantage of their machines; unlikely given their southern base, however. Perhaps TR forms his own party, more viable in divided elections, creating 2 liberal parties and 2 conservative ones.
There's also an easier switch: TR is successful in 1912 and takes the nomination from Taft, ensuring the strength of the Progressives in the Republican Party. This means that the Republicans are not the establishment types like Harding and Coolidge going into the 1920s. The Republicans are urban progressives and the Democrats are agrarian populists. Both increasingly vie for the votes of urban workers, but loyalties split a number of different ways, with difference by industry and by region. If FDR is a Democrat (a big assumption) then, he's a populist. He may end up being a Republican (particularly if he doesn't marry Eleanor) and stay a Progressive. I'm not sure if this is durable, since it requires a very divided American labor movement.
All of this is complicated because Progressivism and Populism are simultaneously very similar and very different: both are liberal reform movements and both favored a larger role for the federal government. Populism is usually strident anti-imperialist/isolationist (a hard sell for FDR) while Progressivism is usually a bit more international. It's tempting to suggest a union between the two movements, but this would prove difficult since neither would bring established machines and both had major issues with the policies of the other.
The biggest question I suppose is whether a two-party system endures past 1896. OTL in 1896 one might suppose multiple parties to be the way of the future, given the success of the Populists and the fragmentation of both parties. If so, then the constitutional reforms of the early 20th century might have made the American system more ammenable: proportional election for some House seats, multiple member constituencies for the Senate, abolition of the electoral college.