As far as I can tell there is no known tune for the one from Richmond given by Bell, = BSP in Margaret Dean-Smith's A Guide to English Folk Song Collections) There are other versions from a variety of locations in England.X:1
T:Poor Old Horse
S:JFSS, #3, 1901, from Sussex
Q:120
L:1/4
M:4/4
K:Fmixolydian
C|F2 (G/A/)B|F3C/ C/|(FE)(F/G/)A|B3(3 G/A/B/|c2Bc|\
F2C2|EFDB,|C3(G/A/)|BABc|(FE2)C|FE(F/G/)A|\
B3B|ccBc|(FE)CE/ E/|(FG)E(C/A,/)|B,2B,B,|(C/D/)EFE|F3|]