I couldn't find this rhyme in some of the most comprehensive collections of nursery rhymes (such as The Oxford Dictionary of Nursery Rhymes and The Annotated Mother Goose). Perhaps it was new to the early 20th century. I found it in The Little Mother Goose (1912), illustrated by Jessie Willcox Smith, and printed in the USA.

If anyone knows anything about the origin of this rhyme, or if you recall hearing it growing up, please email me and let me know. -Mama Lisa