Joe, I agree with you, it's "Mulberry Bush". As Iona & Peter Opie say in The Singing Game (Oxford, 1985, p. 275), "Twentieth-century versions are usually sung to the tune of 'Nuts in May' [i.e., "Mulberry Bush]." The tune is a descendant of "Nancy Dawson", from which other songs, such as "I Saw Three Ships" and "Garden Hymn", are derived.

Almeda Riddle didn't sing it ("Mama, Buy Me a Chiny Doll"; the first version, and the second version, with audio) to that tune; it's a little closer to "Aunt Rhody" than to "Mulberry."

Alic Bertha Gomme's two versions in The Traditional Games of England, Scotland, and Ireland, vol. 1 (1898; Dover, 1964, p. 376) are:

X:1
T:Milking Pails #1
M:C
S:Monton, Lancashire (Miss Dendy); London (A.B. Gomme)
L:1/8
K:Eb
E>E E/F/G A2F2|G2E2F2B,2|E>E E/F/G A2F2|B>G A/F/D E4||
Bc/A/ GFE4|]
w:Last bar only. London version.

X:2
T:Milking Pails #2
M:C
S:Earls Heaton, Yorks. (H. Hardy)
L:1/8
K:A
A A B c d c A2|d c B2c B A2|A A B c d cB2|c2e e c B A2||

~Masato