I remember this cartoon very well. Of course, it was the version that was in an hour-long episode of a Disney TV show that ran from the 1950s to the '70s. It also had a new narrator - Paul Frees, in the character of Ludwig von Drake. But I was fortunate to watch the original version on YouTube.
In this memorable cartoon, the none-too-cheery stories behind well-known nursery rhymes are revealed. It covers "Little Jack Horner," "Mary, Mary, Quite Contrary," and "London Bridge is Falling Down" with a clever contrast of wonderful animation that reflects the influence of the film "Sleeping Beauty" (which was in production then) and serious narration by John Dehner. Each segment begins with an upbeat performance of the song, followed by a depiction of the historical inspiration, and concludes with a rather sombre rendition of the tune.
So overall, I would say this cartoon is that it's artistically beautiful, fun to watch, and really interesting. I especially loved the segment on "Mary, Mary,...", though her story was so sad.
