POD: No U.S. involvement in WWI - either war doesn't occur or it ends quickly.
Without the Red Scare and the division of the Socialist Party over the war and the Russian Revolution, the party continues its pre-1912 growth. The IWW and other revolutionary and socialist currents continue to retain major influence among the labor movement.
In 1916, Charles Evans Hughes defeats Woodrow Wilson and thereafter Republicans continue to hold the White House for several terms. The Democratic Party remains deeply split between a dominant conservative wing and a set of agrarian populists. For about a decade, American politics is deeply fractured, with Republicans dominant and the opposition split between Democrats (primarily in the South), Socialists, and other small splinter parties (Progressives, Farmer-Labor, various independents).
Without the Great War, the precise contours of the Great Depression probably do not occur. But at some point an unregulated Wall Street is likely to blow up and cause another sharp panic. Republicans lose the White House to a resurgent, right-wing Democratic candidate who wins a plurality of the popular vote and an electoral college majority, with an independent candidate backed by a coalition of Socialists, Progressives, progressive Republicans, and populist Democrats outpolling the Republican candidate for second place. The Depression worsens under austerity policies pushed by the Democratic Right.
After one term, this Democrat is unseated not by a Republican but by Norman Thomas, former Socialist leader. The factions that had previously backed the independent fusion candidate in the previous election have now merged into a new "Labor Party," which comes to dominate the Northeast and Midwest - America's industrial heartland, with strong support from some farm states and California as well.
Under a winner-take-all system, the Republican Party fades, unable to retain support except as a residual party in parts of New England and the Upper Midwest. In Congress, the Republicans become a small third-party, typically in coalition with Democrats, and they come to support a common presidential ticket in subsequent elections, known popularly as "the Alliance."
So American politics instead of being Democrats vs. Republicans, becomes Labor vs. Democrats or Labor vs. Alliance.