Here is Mrs Johnstone's tune:

X:1
T:May Song: Bedfordshire
S:Mrs Johnstone, Bedfordshire
N:Learned from her grandmother who came from Carlton
N:Roud 305
Z:Fred Hamer
B:Garners Gay, London: EFDS Publications, 1967, p 33
L:1/8
Q:1/4=100
M:4/4
K:C
G2|c2 c2 G2 G2|E4 C2 C2|A2 A2 G2 F2|
w:Good morn-ing lords and la-dies it is the first of
E6 G2|c2 c2 G2 G2|E4 C2 C2|A2 A2 G2 F2|
w:May, We hope you'll view our gar-land it is so bright and
E6 GG|G2 G2 G2 G2|G4 c3 d|e2 c2 A2 B2|
w:gay, For it is the first of May, oh it is the first of
c6 G2|c2 c2 G2 G2|E4 C2 C2|A2 A2 B2 G2|c6|]
w:May, Re-mem-ber lords and la-dies it is the first of May.