"I'm sure I was singing it before 1971" - Howard, I'm sure you were. I can distinctly recall singing it at Junior School in the UK. The year was probably 1967, although it could have been 1966. A variation I recall is:

Jingle bells, Batman smells, Robin's just as bad.
The batmobile has lost a wheel, the commissioner's going mad.

I don't recall being all that aware of Batman before the Adam West series reached the UK. Certainly the existence of Robin was a revelation to me when I first watched the TV series.