This song can be found in:

John Bell, Rhymes of Northern Bards: Being a Curious Collection of Old and New Songs and Poems, Peculiar to the Counties of Newcastle upon Tyne, Northumberland, and Durham. (Newcastle upon Tyne: John Bell, M. Angus & Son, 1812), page 240.

Only the spelling is different:

About the bush Willy,
About the bee hive,
About the bush Willy,
I'll meet thee alive.

Etc.

Also in

James Orchard Halliwell, ed., The Nursery Rhymes of England: Obtained Principally from Oral Tradition, 2nd edition (London: John Russell Smith, 1843), page 178.