Hi,
This song is in a book which has just come out called 'The Scots Fiddle - Volume 2: Tunes, Tales and Traditions of the Lothians Borders and Ayrshire' by J. Murray Neil. Make sure it's the new one though as this set was originally published as a single volume and has been added to a lot so I'm not sure the one you want would be in the earlier one.
He doesn't give it an author though. The version he gives is from William McGibbon's 'A collection of Scots tunes for the Violin or German Flute' 1762, but he describes it as very ancient. Here are the words he gives for it:

O waly, waly up the bank,
and waly, waly down the brae,
and waly, waly, yon burnside,
where I and my love wont to go!
I lean'd my bank into an aik,
I thocht it was a trusty tree,
but first it bow'd and syne it brak
and sae did my true love to me.

Now Arthur's Seat shall be my bed,
the sheets shall ne'er be pressed by me,
St Anton's Well shall be my drink,
since my true love has forsaken me.
Martinmas wind, when wilt thou blaw,
an shake the green leaves aff the tree?
O gentle death, when wilt thou come?
For o' my life I am weary.

I guess since you're a Scot you'll know where Arthur's Seat is! :-)