Here's the American one: MOUNTAIN DEW, which seems to fit. And there's another version as well: GOOD OLD MOUNTAIN DEW

And here is the Irish song - which I can't believe isn't in the DT, but I couldn't find it anywhere (there is an American version with some lines in common: REAL OLD MOUNTAIN DEW):

THE REAL OLD MOUNTAIN DEW

Let grasses grow and the waters flow
in the free and easy way,
but give me enough of the rare old stuff
that's made near Galway Bay.
The gaugers all, from Donegal,
Sligo and Leitrim too,
Oh we'll give them the slip,
and we'll take a little sip
of the real old mountain dew.
Hi the skiddley-al-the dall
dal the skiddley al-the dee,
dal the dal dal skiddley al the dee,
Hi the skiddley al the dal,
dal mthe skiddley al the dee,
dal the dal dal skiddley al the dee.

At the foot of the hill
there's a neat little still
where the smoke curls up til the sky,
by the whiff of the smell,
you can plainly tell
that there's poitín, boys, close by.
For it fills the air with a perfume rare,
and between both me and you,
as home be go, we will take a little bowl,
or a bucket of the mountain dew.
Hi the skiddley-al-the dall
dal the skiddley al-the dee,
dal the dal dal skiddley al the dee,
Hi the skiddley al the dal,
dal mthe skiddley al the dee,
dal the dal dal skiddley al the dee.

No learned men who use the pen
have writ the praises high
of the sweet poitin from Ireland green,
distilled from wheat and rye.
Sure away with your pills,
it will cure all ills,
be ye Pagan, Christuan or Jew,
so take off your coat, and grease your throat
with the real old mountain dew.
Hi the skiddley-al-the dall
dal the skiddley al-the dee,
dal the dal dal skiddley al the dee,
Hi the skiddley al the dal,
dal mthe skiddley al the dee,
dal the dal dal skiddley al the dee.

The chorus varies - everyone seems to do the lilt a bit different (skiddery al, or dithery dal, or skiddley al etc.

And in Colm O Lochlann's Irish Street Ballads he writes in the notes: "I first heard this song at a meeting of newly released political prisoners in December 1916. I am told it was written by Phil O Neill of Kinsale."