Perhaps not Alexander's tune, unless it was written quite a long time before he published it (Lady Nairne died in 1845 at the age of 79). 'Tramps and Hawkers' is unsuitable metrically.

Many of Lady Nairne's songs can be found in older collections of popular Scottish music, and that would be your best bet for notation, I expect; however, there is a cd by Anne Lorne Gillies, White Rose o' June -The songs of Carolina Oliphant (Lady Nairne) (Brigh Productions Dunlop BR002) which includes arrangements of both the songs you have asked about. Presumably the tunes used are the ones Lady Nairne intended.

Incidentally, there is a transcription of Songs by the Baroness Nairne (1902 edition) at http://www.electricscotland.com/nairne/; tune indications are not given.