Here is a satire pastiche from May 1939,
The Stick
by Fitchburg State Teachers College

Page -6-

There were three men in bed and the middle man said, "Roll over." So they all rolled over and Dick Kelliher fell out.

This is a 1945 report of a Scouts concert... . Ten in a Bed.” illustrated the 3rd Ballymena (West Church) Scouts’ idea of solving the housing problem.

There are hundreds if not thousands of nursery/song books, teaching aids, cd's, Lps where the song is listed as Public Domain/Trad. or not credited as anything at all.   The Copyright Societies list it as such, and as far as can be told, anyone can use it, or make their own claims on arrangements. Apart from nursery use, it is reproduced as a camp/Scout song in many places. There are no traceable claims, that would indicate a copyright issue.