Here is an interesting study
of how lyrics
can change when passed down via an oral tradition
Compare RandomIdiot's version with the playground version that we used to sing here in Sydney, Australia circa 1978:
Jingle Bells, Batman smells
Robin ran away
Wonder Woman lost her bosoms
flying TAA.. Hey!
As far as I can remember there was no second verse, the first verse was repeated ad infinitum.
As stated in a previous node, Jingle Bells was written in 1857. Batman first appeared in Detective Comics in 1939. The Batman television show appeared in 1966, and was certainly being shown as re-runs in Australia around 1978, when the above lyrics were current.
Wonder Woman first appeared in All Star Comics no. 8 in 1941. There was a Wonder Woman TV show, also shown in the 1970's.
"TAA" was the acronym for the now-defunct domestic carrier "Trans-Australia Airlines", which operated from 1946 to the 1980's
Bosoms have been around since the first mammals started appearing about 200 million years ago.
I'd speculate that the version I've reproduced, above, came about from Aussie kids watching the two TV shows interspersed with TAA's "fly the friendly skies" commercials. It's possible that an adult came up with the lyrics, though - "bosoms" is not necessarily in most five-year-old's vocabularies - I think I learnt the meaning of "bosoms" from this rhyme!
Why Superman is not mentioned is an unanswerable question. The George Reeves Superman TV show predates both the Batman & Wonder Woman shows.
Readers will be happy to note, that although the standard "Jingle Bells" is usually reserved for Christmas, the "Batman smells" version can be enjoyed all year round.