I can only speak about the European part of things, specifically Austria as that is my home country: Both Germany and Austria went through a process called "Entnazifizierung" (Denazification) which prohibited a lot of former officials from holding any kinds of government positions in the future. This was a pretty lengthy process as quite a large part of the popluation had been involved in national socialist affairs and you couldn't just remove every government official. Many will still claim today that the process was not thorough enough as there absolutely were former Nazis holding positions after the war, but please remember that manpower was in dire need after the war after losing such a big part of the population and it's almost impossible to distinguish between a true Nazi and a collabarateur.

That being said both countries had quite a few politicians available that were not involved in the Nazi regime, those being the ones that had spent the last few years imprisoned or in Concentration camps. For Austria two examples would be Leopold Figl as well as Karl Renner, one a Conservative, the other one a Social Democrat. Figl was brought to a concentration camp in 1938, Renner had been under house arrest throughout the war. Figl ended up as the first Austrian chancellor whilst Renner became the first President.

Please also remember that both countries remained officially occupied for a while after the war, with Austria losing its status as an occupied country in 1955 when we declared neutrality.

Unfortunately I cannot really provide you with unified sources as my information comes both from information I read over the years as well as the information I got from my parents and grandparents that lived through that time.