From Allan of Australia, writing in 1999

Professor Child, writing in 1890 (or earlier) says that "Waly, Waly" was known from the first quarter of the eighteenth century. This in his notes on "Jamie Douglas" (Child #204), where he points out the common ground between the former and several versions of the latter.

Bronson says that Allan Ramsay was the first to publish it in Tea-Table Miscellany, c. 1724.

While searching for something else in the Bodleian Library Broadside collection, I came across a broadside from 1813 which contains "Waly, Waly" and attributes it to Allan Ramsay in the "Tea-Table Miscellany"

the Bodleian catalogue details are as follows

Sheet Title:         The thistle
Printer:         Catnach, J. (London)
Date:         between 1813 and 1838
        Imprint: Printed by J. Catnach, 2, Monmouth-court, 7 Dials
Illus. Ballads on sheet: 17
        
Copies:         Johnson Ballads fol. 28

The Broadside may be found here

Not conclusive evidence, but indicative.