The original theme was much more interesting ... but this subject has been hotly debated for 60 years now, with claim/counter-claim ad infinitum. My own belief is that he had a full set. Why? Because Little Eva, being a Frau, would have been unable to keep as juicy a secret as that from her girlfriends, and it would have been all over Germany even before the war. AFAIK there is no evidence to suggest that it was (open to correction, as ever).
My Cockney Granddad used to sing that and other such stuff when I was a kid. And he should know, like many he stood up to the Nazi fcuker toe to toe and kicked him in the only good one. Well done Granddad. R.I.P.