We got a book of the children’s songs and at least one out of three are set to this tune. Jane Taylor probably wrote the poem, but that is not the same as writing a song. Otherwise The Byrds’ would get no credit for “Turn, Turn, Turn”
Also, this link states that the tune was first published in 1761 in France but goes on to say that no one is sure of its exact origins. I would say that the correct answer to who wrote it is, “We don’t know.”