Version from Beth's Notes. She credits Opie's "The Singing Game" as the source. I think this version is from Sail Away.
Recordings from the Lomax Digital Archive that mention "Sally Go Round..."
Eddie Nelson, 1937 - South Carolina
This melody is not notated in any source I've found.
Dorothy Mae Woods, 1942 - Mississippi
The entire piece is performed on one note.
Unidentified woman from Mississippi, 1942
Performed more like a chant than sung on one note.
Performed to the melody of "Here We Go Looby Loo," - see the version from Opie and Opie's The Singing Game below.
"Sally Go Round the Sun" from Sail Away. The song is credited as a "Black children's game song from St. Helena's Island" but no actual sources were cited.
Version from Opie and Opie's "The Singing Game." This one appears to have been conflated with the melody of "Here we go Looby Loo."
Version (below) from Old Surrey Singing Games & Skipping-rope Rhymes by Alice Gillington, 1909.
Version from Old English and American Games by Florence Warren Brown and Neva Boyd, 1915.
Version from 1898's 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 · Volume 2 by Alice Bertha Gomme. The tune is not notated.