Weirdly enough, Cracked.com seems to have the most rigorously researched history of the song. There have been a number of people who remembered it coming up in the mid-60s in California, possibly partially as a result of the Batman TV show, but the earliest citation is 1967 from the Lawton Constitution, source being an Army child in Belgium.
"American kiddos in Europe aren't completely divorced of the U.S. gimmicked tunes. Li'l Jana Montgomery, daughter of Maj. and Mrs. Ross D. Montgomery, formerly stationed at Fort Sil and now with MAAG headquarters in Brussels, Belgium, warbled this tune during the holidays:
"Jingle bells, Batman smells;
Robin rang away;
Batmobile lost a wheel -- and
Commissioner's stuck in sleigh."
As Clara Díaz Sanchez notes in a comment, Tom Scott did a survey of the remembered lyrics, and found not only a very wide variation of answers, but also that there is a distinct variation in the second line being "Robin laid an egg" or "Robin flew away" between the United States and the United Kingdom. He did not find direct evidence for his hypothesis that the appearance of the song in an episode of The Simpsons influenced the variation, but the statistics are suggestive of it.
Lastly, as others have noted, it looks like the first acknowledgement from DC Comics of the song was Joker singing it in the Batman the Animated Series episode "Christmas with the Joker".