I can't pinpoint when I first heard this rhyme, but it certainly seems to be well known in the USA since at least the 1990s.

Here's some information from http://en.wikipedia.org/wiki/Jingle_Bells

A well-known children's playground song uses the "Jingle Bells" melody of the parody of Batman:

Jingle bells, Batman smells,
Robin laid an egg;
The Batmobile lost a wheel,
And the Joker got away! (Joker did ballet Or "Batman did ballet!" usually with a shout of 'Hey' at the end)

This particular version was sung (by the Joker himself - as voiced by Mark Hamill) in "Christmas with the Joker", an episode of Batman: The Animated Series. The Joker also adds the line while escaping Arkhum Asylum:

Crashing through the roof,
In a one horse open tree (the Joker was using a rocket powered tree).
Busting out I go.
Laughing all the wee-he-he

-snip-

The Wikipedia page for the "Batman" tlevision program http://en.wikipedia.org/wiki/Christmas_with_the_Joker#Season_One_.281992-1993.29 gives November 13, 1992 the date for that particular episode.

For what it's worth, I've never heard or read any children, teens, or adults recite the Joker's version given above.