I'd forgotten the Oswald link. There were two 18th century tunes. One (the one associated with Rutherford's song?) was much further removed from the 17th century tune.

The title "Flowers of the Forest" is right there in the Skene Manuscript. But there is an intermediate version of the tune predating Oswald where the title had been lost:

X:3
T:McDonalds Rant
S:Gairdyn MS (1723), NLS MS.3298
N:Dorian mode in MS but sounds better in major/mixolydian (has 7th gap)
M:4/4
L:1/8
K:Dmix
FA|:A2B2 defd|gfed BdBA|FAAB defd|(ga)(fg) e2d2:|
a2a2 fgaf|g2g2 gabg|a2a2 fgaf| gafg e2d2:|

So, the tune survived through the years between the Skene MS and Oswald without any associated story about Flodden. Which suggests to me that there was no story about it to begin with.

I had no compunction about changing the mode from the way it is in the MS, since that MS was obviously written as an aide-memoire with no thought of anybody but the compiler using it. A lot of the tunes are fragmentary or near-gibberish.