The scene I'm working on takes place in West Germany in 1983. The speaker is a 33-year-old German woman, born, obviously, after the end of World War II. She's telling my protagonists the story of someone she grew up with, and the sentence in question goes, "His father fled to Switzerland during the war because he did not want to serve in the army for the Nazis." But I strongly suspect she would not use the N-z term. Am I right? If so, can anyone tell me what term would have been/would be used by those who repudiate the horrors of the time, when referring to those who perpetrated them? NOTE: She wouldn't say "the German army" because that would imply the father was unpatriotic, and she doesn't want to give a bad impression of him.
That National Socialist Party ("the Nazis") didn't control the army, except for a few dedicated regiments. But the speaker could very well have referred to "Hitler's army," since he was personally in control of the entire military apparatus by the time the war was underway. There were many people who were dismayed at the way Hitler was removing the old leadership of the army and replacing it with his own cadre of loyalists. It was this sentiment, as much as anything, that drove these "old guard" people into attempting to assassinate him in 1944.
That's not entirely true. The Waffen-SS numbered nearly a million men and was under the control of Himmler. Although this was a lot smaller than the Heer, they were nonetheless more significant that just a few regiments. Going back to the OP, the Germans had been de-Nazified after the war, and I suspect she would use the term, as a matter of pride, if the father fled rather than served them. The WW2 German army was called the Heer, which just means "Army". Before 1912 and after 1956, it was referred to as the Deutsches Heer (German Army). The overall German WW2 armed forces was called the Wehrmacht, which included the Kreigsmarine (Navy) and the Luftwaffe (air force). The World At War contains some interviews with WW2 German general and others, and I don't *recall* any of them not using the word, but it's been a while since I watched it. At least one ex-WW2 general (Hans Speidel) reached a high rank in NATO.
I am hoping to see a German friend later in the week (lockdown rules permitting) and will ask her - 2nd generation so was born shortly after the end of WWII but her mother lived through. My understanding is that "the Nazi party" was just a political statement at the time, so no more damning than "democrat" or "republican" so it would have been fine to say that the character didn't want to work for the Nazis. The meaning of the word has only subsequently been changed virtually to profanity.
The army may not have been made up entirely of Nazi members, but surely they controlled it, given that they were the ruling party of the country?
It's complicated. There was a great deal of rivalry and power jockeying among generals and political figures, plus the distinction between the regular army and the SS, as already noted by @Naomasa298. The original question, I think, has already been adequately answered. The character could say "Hitler's Army" or "the Wehrmacht" or even "serve in the army for the Nazi regime" and they'd all make sense conversationally.
Honestly, I think you could completely sidestep this problem by opting for "His father fled to Switzerland during the war because he did not want to serve in the army." The context of a German woman in 1980s West Germany saying this about another German's father makes it quite clear which army and war she's talking about, which in turn heavily implies why he didn't want to serve.