POD: Leon Czolgosz gets a cold and stays home. William McKinley is not killed. Leads to a TL something like this:
Teddy Roosevelt is denied the GOP noimination in 1904. William Jennings Bryan wins the 1904 elections. TR runs as a joint Republican-Progressive Party candidate in 1908.
In 1916, TR runs for a third term due to WWI (not quite the same as OTL). The Republicans (i.e. Old Guard Conservatives) defect and run a protest candidate. TR still manages a near win.
In 1920, Republican candidate wins the White House.
In 1924, Democrat wins. Re-elected in 1928.
In 1932, Progressive party wins in a landslide. Socialist Party also sees marked gains. Republicans and Democrats are increasingly conservative, though of different stripes.
In the 1940s and 1950s, Republicans and Democrats run in alliance against Progressive and Socialists. Socialists lose ground due to fear of communism in the Union of Asian Peoples' Republics (India and China).
In the 1960s, oppoisition to Civil Rights legislation prompts official union of Democratic-Republican Party (marketing itself as the Party of Jefferson). Socialists remain a third party. Progressive Party in power.
Due to hung election in 1968, electoral college abolished in favor of a system resembling the Birch-Bayh Amendment.
In the 1970s, social conservatism allows Dem-Reps to gain power for first time since merger.
In the 1990s, Progressive Party adopts "New Centrism."
In 2008, Socialist Party elects first POTUS.