Robert Burns is quoted in James Johnson's Scots Musical Museum as saying that the words of The Boatie Rows were written by a John Ewen (ca. 1761-1821) of Aberdeen. The lyrics appear in Volume 5 with three different tunes (pp. 437-439, song nos. 425-427). Here's what Burns wrote in the notes: "The first four bars of the first air, No. 425, are taken from the tune of Weel may the keel row, and the rest from the tune of There’s nae luck about the house.” Burn goes on to state that the second tune “has never become a favourite with those who sing the ballad,” and writes of the third tune: “This fine modern air is the genuine tune of the ballad. Some years ago it was arranged as a glee, for three voices, by William Knyvett of London, and has deservedly become very popular.”

So the song's not entirely by Anon. Trad. after all.