Let’s clear this up.

There are three distinct tunes associated with The Braes o’ Balquhither.

A) The commonly accepted melody used by Burns for ‘An I’ll Kiss Ye Yet’ and printed by R A Smith under the title The Braes o’ Balquhither, Smith’s Scotish (sic) Minstrel, Vol. 1, 1822.
B) The Three Carles o’ Buchanan, the tune chosen by Tannahill in his letter to Smith, 1810.
C) The Wild Mountain Thyme, first recorded by the McPeakes in 1956.

All tunes are COMPLETELY different.

The tune Rory referenced, New York publication, is tune B also published by Smith Vol lV 1824 2nd edition, titled The Three Carles o’ Buchanan and not the same as Daniel referenced (A).

Tune C is completely original and credited to the McPeakes.

Daniel, John Hamilton was know to Tannahill so he probably deliberately chose the tune tune (B) to distance himself from Hamilton’s Braes o Bowhether which has a familiar theme. Hamilton, by the way, copies many lines from Burns’ An I’ll Kiss Ye Yet, including the tune (A).m

I am an ethnomusicologist, degree from the university of Edinburgh, 2008, Thesis title “A Study of Two Tunes: The Three Carles O’ Buchanan and the Braes O’ Balquhither in their cultural contexts from 1740 to the present day”.

Daniel, I suggest you listen again to tunes A and B and take note of the differences.
Rory, both your New York referenced tune (B) and the one you referenced earlier form the Scotish (sic) Minstrel (A) are different.