I have a copy I xeroxed from a different book, but have unfortunately lost track of the title of it. The text is the same. The song is said to come from 'Sawney's Letters, or, Cariboo Rhymes. From 1864 to 1868', which is perhaps a slightly later reprint. I don't know if the tune was given in the early publications. The tune for the copy I xeroxed is given as follows:

X:1
T:Bonnie Are The Hurdies, O!
Q:1/4=100
L:1/8
M:6/8
K:Bm
E|"D"F2FF2E|D2DD2F|"Em"B2BB2B|B2EE2F|\
"G"G2GG2B|"D"A2FD2F|"Em"E2GF2E|"Bm"DB,2B,3||\
"D"FF2F2E|D2DD2F|"Em"B2BB2B|B2EE2F|\
"G"G2GG2B|"D"A2FD2F|"Em"E2GF2E|"Bm"DB,2B,2||]

There is no song of 1549 related to Burns' song. There is only the title "Cou thou to me the raschis grene" in 'The Complaynt of Scotland', 1549. The only songs that fits that title is an English one "Colle to me the Rysshys grene" given (with music) in Ritson's 'English Songs' and (without music) in 'Captain Cox, his Ballads and Books', Ballad Society, 1890. It has nothing in common with Burns' song, and is to a different tune.