According to James Dick in 'The Songs of Robert Burns' p. 464, the tune below is a version of "Put in All" in 'Pills'. Since the original of "Ye Jacobites" that I cited from a chapbook was to "Captain Kid", then the original "Chimney Sweep = Coming down/ Captain Kidd" is most likely this tune. What is probably the definitive work on this is an article I don't have, Bertrand Bronson's 'Sam Hall's Family Tree' in 'The Ballad as Song', 1969. Steve Roud's broadside index lists several copies of "Admiral Benbow" (note that there are two that go by this and other titles), but none contains a tune direction.

X:1
T:Ye Jacobites by Name
N:'Scots Musical Museum', #371 (1792)
L:1/8
M:C
K:Am
E2|A3/2 B/ A G E2 E E|c2B A {A}B3G|\
A3/2 B/ A G E2 E3/2 ^G/| A6||(AB)|\
c3/2 d/ e f g3(f/e/)|d c B A G2(cd)|\
e A B A (E3/2D/) E ^G|A6|]