I have a printed music sheet dated 1810 of Tannahill's Braes O' Balquhither to the tune he set it to, The Three Carles o Buchannan. This is different to the 'normal' tune which Smith set it to in 1824.
I researched this about 12 years ago and found a similar, Scottish Gaelic tune called Brochan Buirn. Ms Gilchrist the musicologist said this was the tune her mother sang to The Braes and that it was different to the 'normal' melody.
It is very similar to 'The Island of St Helena'
John Moulden printed Hugh McWilliams 'The Lass Among the Heather' tune Saint Helena. It appeared in a collection of his poems in 1831.
The lyrics of this song were conflated with Tannahill's Braes by Jeannie Robertson but she called it the Braes and her melody, when all the ornamentation is stripped away is a close variant of 'The Three Carls o Buchannan.
I would suggest the St Helena tune is Tannahill's Three Carles and McWilliams poem, hence Jeannie Robertson, was sung to this tune which was used by "Bonnie's Awa" thus McWilliam gave his song tune title as St Helena.
This is a very short summary of my research.