I've never heard it sing to the tune I associate with Villikens. The tune I'm familiar with is distinctly different - though of course the meter is the same, so I imagine both tunes may have been used for both songs.

Colm O Lochlann in his notes to Master McGrath in "Dublin Street Ballads" writes "Often heard in Dublin to the same tune as 'The game played in Erin/go/bragh'...Also sung to 'Villikens and his Dinah'."