Chappell (Popular Music of the Olden Time, I, 1859, p.152, comments:

"The correct date of this fine old melody appears altogether uncertain, as it is to be found in different forms at different periods; but it is here placed in juxtaposition to Sir Edward Noel's Delight, and All you that love good fellows, or The London Prentice, because evidently derived from the same source. The commencement of the air is also rather like Prince Rupert's March, and the end resembles Old King Cole, with the difference of being major instead of minor."

Claude M. Simpson (The British Broadside Ballad and Its Music, 1966, pp. 13-16) discusses All you that love good fellows in some detail, including its association with The Honour of a London Prentice, and its appearance under the names Nowells Delighte (Welde MS Lute Book, c.1600), Nancie (Fitzwilliam Virginal Book, arranged by Thomas Morley), and so on.

X:1
T:A true discou[r]se of the winning of the towne of Berk...1601.
S:Shirburn MS, 1585-1616
B:Claude M. Simpson, The British Broadside Ballad and Its Music, 1966, p.14
N:Original is unbarred.
L:1/8
Q:1/4=100
M:4/4
K:C
z4 G4|c2 G2 c2 d2|e4 d4|c2 G2 GFED|
C8|z4 G4|c2 G2 c2 d2|e4 d4|c2 G2 GFED|
C8|d4 d2 d2|d6 d2|d2 c2 B2 A2|G4 e4|
d2 B2 d2 c2|B2 G2 G2 A2|G3 F E2 D2|c8|]

The tune was taken up by a number of 17th century Dutch songbooks, so it was current outside England from that time.

X:2
T:Sir Eduward Nouwells Delight
S:J.J. Starter, Friesche Lust-Hof, 2nd edition, 1621.
B:Claude M. Simpson, The British Broadside Ballad and Its Music, 1966, p.15
N:Original is unbarred and, though pitched in C, contains key signature of one sharp.
L:1/8
Q:1/4=100
M:4/4
K:C
c2|c2 G2 c2 d2|e2 e2 d2 B2|c2 G2 GFED|C6 G2|
c2 G2 c2 d2|e2 e2 d2 B2|c2 G2 GFED|C6 c2|
d2 e2 d2 A2|d2 d2 d2 e2|d3 c B2 A2|G2 G2 a3 f|
e2 c2 e2 d2|c2 G2 G2 G2|(AB) c2 c2 B2|c2 c4 G2|
A2 G2 A3 G|A2 G2 A3 G|F4 E4|D6 c2|
d2 e2 d2 c2|d3 e d2 (gf)|e2 (dc) d3 c|c6|]

"The British Grenadiers", says Simpson, "clearly descends from this old tune."