Oh, correction: The tune I know is clearly not by Stevenson and is not part of the poem itself. The record (though not the original pressing) is listed at
https://www.discogs.com/Seymour-Barab-Russell-Oberlin-A-Childs-Garden-Of-Verses/release/9969296. Here's the review I've just put there:
This is a wonderful recording, but this is not the original release. I well remember listening to it in my bedroom as a child, probably in the late fifties, and wondering at the strange label name, "Esoteric", which was indeed esoteric to me. The cover was pink and didn't have Steig's art. I have a very strong memory for songs in general, and I still know and sing some of these, including The Swing, Windy Nights, and Where Go the Boats.
And I've added this:
PS: The original is listed at
https://search.library.wisc.edu/catalog/9910128657402121 , which says "Publication New York : Esoteric Records, [1953]"