I asked Tim about the tune he used. His comment was that it "came from the deepest recesses of my head!" i.e. he neither consciously borrowed it nor consciously composed it. Brendan Begley reckons the first line is familiar from an Irish language song - but the second part is different.
FWIW, I reckon the fist line bears more than a passing resemblance to that of An BinsÃn Luachra / Bonny Bunch of Roses.