This is the song from Thomas Ravenscroft's Deuteromelia, published in 1609. Ravenscroft was publishing songs sung in taverns during the Elizabethan period, which were falling out of favour under the new king, James I. That would suggest this was being sung about the mid 1500s, and possibly earlier. I have retained Ravenscroft's spelling, which dates from an age when people had a more relaxed attitude to that sort of thing. I find the suggestion of a connection to the Barley Mow reasonably convincing:

Give us once a drink for and the black bole
Sing gentle Butler balla moy
For and the black bole,
Sing gentle Butler balla moy

Give us once a drink for and the pint pot
Sing gentle Butler balla moy
The pint pot, for and the black bole
Sing gentle Butler balla moy

Give us once a drink for and the quart pot
Sing gentle Butler balla moy
The quart pot, the pint pot, for and the black bole
Sing gentle Butler balla moy

Give us once a drink for and the pottle pot
Sing gentle Butler balla moy
The pottle pot, the quart pot,
The pint pot, for and the black bole
Sing gentle Butler balla moy

Give us once a drink for and the gallon pot
Sing gentle Butler balla moy
The gallon pot, the pottle pot,
The quart pot, the pint pot,
For and the black bole
Sing gentle Butler balla moy

And so on, until:

Give us once a drink for and the tunne
Sing gentle Butler balla moy
The tunne, the butt
The pipe, the hogshead
The barrel, the kilderkin
The verkin, the gallon pot
The pottle pot, the quart pot,
The pint pot, for and the black bole
Sing gentle Butler balla moy

Note it is the black bowl here - the basic mediaeval drinking vessel was of leather waterproofed with pitch. This would presumably predate the brown bowl of glazed earthenware. Ravenscroft prints the 'Balla Moy' of the chorus in italics, perhaps to indicate that he doesn't understand it but is just printing what he heard - it could be a mondegreen for 'Barley Mow', which may well have meant nothing to a city dweller like Ravenscroft.

Or maybe our 'Barley Mow' developed separately, without reference to the 'Balla Moy' - what do you think?

Andrew