I can only speak for Denmark.

The DNSAP (Danish National Socialist Workers Party) was completely ousted from all political offices in 1943 and onwards. During the german occupation, Denmark still held elections, and it wasn't until august 1943 when the government collectively stood down in protest, that the germans occupied the country de jure. As for the DNSAP, the german administrator, Werner Best, didn't trust them, and thus kept them out of political influence despite their SS and Nazi leanings.

After the war, the party itself wasn't outlawed, and still maintains a very, very fringe presence today. It does not have any representatives in the parliament or in municipality councils.

Sources: A Story About Denmark: Lidegaard, Martin The Jewish Action, 1943: A collection of documents from the peoples involved, including the kriegsmarine, DNSAP, Thorvald Stauning, Werner Best and various members of the resistance.

One obvious thing to point out is that communist one-party states were established across eastern Europe after WWII.

This was especially important for Poland because Polish disdain for both fascism and communism, which was largely due to the way both the Nazis and the Soviets had treated Poland during the war. This led to a lot of political tension. Regardless, the communists held a controlled election and were able to establish superiority in the government. In the end, non-communist political parties were entirely abolished from the new Polish state, leaving the Polish United Workers' Party in charge.

This would include fascist parties in Poland, which were treated with vigorous prejudice by the communists for their role in the Second World War. Many Workers' Party officials were removed from their positions (or even executed) off of suspicion that they were expressing 'nationalist' tendencies (read: fascism, which is an ultranationalist movement).