This is a tough one.
The following from James J. Fuld, The Book of World-Famous Music.

"The earliest printing of this song was probably about 1840-1841, when it was published in sheet music form under the title Bonnie Loch Loman. The cover states that the song is from "Vocal Melodies of Scotland," and is arranged with symphonies and accompaniments for the piano forte by Finlay Dun and John Thomson...there is a reference to Stationers' Hall; ...states written by a Lady. Arranged by Finlay Dun. ...The back cover advertises contents of the four volumes of the "Vocal Melodies of Scotland.".... The song was not in fact included in any of the four volumes of "Vocal Melodies of Scotland," at NLS [National Library of Scotland], nor entered at Stationers' Hall."

Fuld continues his discussion, comparing some melodic phrases with "The Bonniest Lass in a' the World"...[1733].... "However in this writer's opinion [Fuld], "Loch Lomond is sufficiently different to be considered a distinct song."

Dang near everybody who has commented on Scottish song has suggested authors, i. e. Lady Scott, or tried to tie it to the times of Bonnie Prince Charlie, but nothing convincing.

Jack Campin (above) lists a tune from Olson that is very similar to the tune of "Loch Lomond," but for lyrics, "unknown" seems to be the most accurate assessment.