Wild mass guessing.

In one of their uncommon declarations of canon facts, Valve has stated that the Medic of Team Fortress 2 is not, and was never, a Nazi. Quite understandable, both to avoid controversy and because it'd be too easy and boring to go that route.

This leads me to start pondering things.

Another declaration of canon is that the game takes place in 1968. Assuming Medic is the oldest of the group, he'd be in his late 40s or early 50s - I tend to say 48 or 49 for conveninence, so he'd have been born in the late 1910s or early 1920s. Again, for convenience, I tend to go with 1920. He was also raised in Stuttgart. Also keep in mind that while joining the Nazi party itself wasn't mandatory for adults, joining the Hitler Youth was made so for children in 1936. I can't see how, with the political climate as it was and the social pressures as they were, Medic would have been able to avoid joining the political party and stay in Germany as a accepted member of society.

I've seen it suggested somewhere that he left the country for the duration, which makes quite a bit of sense and would be eminently understandable given both the real world's history and the game's worldbuilding. Some days I go with this, but for some reason on other days I go with something else. Mostly because I know that if it was handled properly, Medic having a camp tattoo would be incredibly compelling.