Folk tune ID - "I Was Not A Nazi Polka"

There is this satirical song by The Mitchell Trio from 1965 entitled "I Was Not A Nazi Polka":

The lyrics start with

As you travel through Die Schöne Deutschland
A melody will greet your ears
It's a melody that's been around in Deutschland
For fifteen to twenty years

I know that the words refer to the political-social situation of that time- but I am interested in knowing what is that melody that they use in the beginning? I thought it was an original composition, but then after years of listening to it I heard it also being sung / vocalized in some feature movie, which action took place in Denmark in 1875

So what is that melody that's been around in Deutschland and its north neighbor for supposedly at least 150 years? Does anybody know?

According to

"I was not a Nazi Polka" could be a parody of "Twelve days of christmas".

I have no proof if this is true.

Or maybe you read this thread with some other suggestions.

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That thread says that the lyrics [concerning the political satire] were supposedly written by Norman Martin. And the the song incorporates traditional German drinking song "Shnitzlebank" [and it does sound like it as far as I can tell from what is available on YouTube]

And there are some ideas about that original first tune, but unfortunately with false answers