The situation on the left would be pretty simple, because it exists in nearly all first-world PR democracies, and to a large extent existed in the Democratic primary in the last 4 months of 2015, between when Sanders became a serious representative of the left and when the polls started giving him a real shot at winning the primary. There's a mainstream center-left party, which is in a state of constant compromise between various factions - organized labor, educated secular liberals (e.g. the pro-choice movement), blacks, immigrants. There are parties to its left, which attempt to pull it leftward on a host of economic and social issues; in Europe's PR states, there's typically a green party that's completely mainstream and enters social democratic coalitions as a junior partner, and a left-populist or communist party that's not yet mainstream and at best supports the social democrats' coalitions from the outside.
I find it unlikely that there would be real fracturing on the left in the US. Blacks would not form their own party, unless you're changing the entirety of civil rights history - African-American politics is integrationist. They'd be a major part of the Democratic Party, with almost no defections to the Greens or the Socialists. Sanders' own socialism is based on rural populist traditions and is almost entirely white. The Democrats might well still refuse to use the phrase social democracy to describe their platform, and stick to calling it liberalism or progressivism.
The center and right are where all manners of things could happen. This is for two reasons:
1. PR democracies are split between ones with a dominant center-right party, such as Germany's CDU, and ones with two poles, such as Belgium and the Netherlands' Christian Democratic and Liberal parties. The latter kind tend toward tripolarism between social democrats, liberals, and Christian democrats, rather than left-right bipolarism, and coalitions require two out of three.
2. Far-right populism is challenging the system in Europe. The extent to which the center-right considers the far right legitimate strongly depends on the country and the center-right leadership, and the whole system is in flux. In the US, the Dixiecrats have always existed, and would form their own pole, without needing to pretend to be Democrats or Republicans.
The result is that any of several arrangements is plausible:
- There's a Southern racist party, which has the balance of power and plays the Democrats (and allies) and the Republicans (and allies) against each other to maintain white power. This is essentially OTL in the 1950s.
- There's a Southern racist party, which the other parties consider illegitimate, leading to grand coalitions in Congress between Democrats and Republicans. The Greens and Socialists are constant critics of this arrangement from the left, calling the Democrats atrophying elites.
- The center and right consist of two parties - the pro-business Republicans, and the Christian Party taking in Evangelicals and rural populists. They're allies against the Democrats. They both have internal tensions regarding issues of race and immigration.
- The center and right consist of two parties - the Republicans, and the Christian Party. But they're not consistently allies. The Democrats often ally with one of the two, working on anti-poverty legislation together with the Christians and on LGBT rights and civil rights with the Republicans.