Version from Beth's Music Notes. A version without sheet music is available on Mama Lisa's with absolutely no sourcing.
The above version can be found in Jill Trinka's "My Little Rooster." (Trinka, J. (2007). My Little Rooster. GIA Publications, Inc. )
Quaver Music also has a version.
This blog claims it can be found in Ruth Crawford Seeger's American Folk Songs for Children, but this is not correct. This blog claims it is from 120 Singing Games and Dances for Children and that is ALSO not correct.
This video from "The Clancy Brothers and Tommy Makem In Person at Carnegie Hall" has the song at 5:40. The Clancy Brothers were an Irish folk music group that started in the 1960s.
The River of Song blog follows the claims of the Clancy Brothers, saying it is a game played in Tipperary, Ireland.
Unable to find an authentic folk source with a melody - possibly another composed piece like "Rocky Mountain." Found similar words to modern version in Herald of the Star.
MacGregor, Theodora. "Visit to a Play-Centre, Herald of the Star, Vol. 7 (1918). United Kingdom: 603-607.
The below version is found in a collection of "Ring Around the Rosie"-type games and was collected around 1898.
Gomme, A. B. (1898). The Traditional Games of England, Scotland, and Ireland: With Tunes, Singing-rhymes, and Methods of Playing According to the Variants Extant and Recorded in Different Parts of the Kingdom. United Kingdom: D. Nutt. p. 110
Another source with different lyrics:
The leaves are green, the nuts are brown
They hang so high they won't come down
Leave them alone 'till frosty weather
And then they'll all fall down together
Gavin, H. "From Old Students." The Child Life Quarterly. (1900). United Kingdom: George Philip & Son. pp. 122-123
Further research shows that the above words could be sung to the tune of "Nuts in May" which is very different than "Frosty Weather."
Thoyts, Emma Elizabeth. "Old Berkshire School Games." The Antiquary. (1893). United Kingdom: E. Stock, pp. 253-255.