You will be unlikely to get anything more definitive on this song than the information posted by John Moulden in a discussion here a couple of years ago:Lyr Add: THE BONNIE BLUE-EYED LASSIE
It appears that the only known traditional source for the tune was Elizabeth Cronin, and that the song has been found once only in tradition elsewhere (in Scotland, early 19th century, no tune recorded). Of course, it has been widely recorded commercially in recent years, often with the words considerably altered, for one reason or another.
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-Joe Offer-