Here is the tune as given in The Dubliners' Songbook:

X:7
T:The Twang Man
S:The Dubliners' Song Book, Music Sales Limited, 1974
M:6/8
L:1/8
K:D
D3 | F2 F (FE) D | A3 B B, B, | (B,A,) B, D2 D | D3 A A A |
w: Come list-en to_ me sto-ry, 'Tis a-bout_ a nice young man, When the Mil-
B2 B (BA) F | (d2 e) d2 B | A2 F FE D | E3-E2 A |
w: ee-tia was_-n't want_-in' He dealt in hawk_-ing twang._ He
B2 B B A F | (d2 e) (dc) B | A2 F (FE) D | E3 D2 E |
w: loved a love-ly mai-den_ As_ fair as a-ny__ midge, An' she
F2 F (FE) D | {D}A3 HB2 B, | B,>A, B, D2 D | D3 |]
w: kep' a tray_-cle de-pot Wan side of the Car-lisle Bridge.

The notes in the book says:

A rare bit of old Dublin from a broadsheet ballad. The tune is Limerick Is Beautiful.

I don't know "Limerick Is Beautiful", but this doesn't sound much like what Ronnie Drew sings on the record.