Nonsense, Bee.
The song is in editions of Mother Goose and other collections of nursery rhymes of England, and has been for two centuries.
See S. Baring-Gould, "A Book of Nursery Songs and Rhymes," XXVI, p. 39, among others.
Iona and Peter Opie, "The Oxford Book of Nursery Rhymes," p. 186, includes The Carrion Crow.
The rhyme is known in New England and across America and Canada.

The chorus is found in old editions and a copy in the Bodleian Library (ca 1800-1820) has the chorus; whether it is found with the first known copies I don't know.