The lilting bit is scatting, written phonetically as:
Skiddly idly diddly idle, etc,
and the unknown river is probably the 'Banks of the Lough' (pronounced lock, with a soft k), which could be a tune, although not one I'm familiar with.
And Pete M -the playing the jigs and reels came after at the party, because there is no way on earth you can get an Irish person to keep quiet at a party!! It probably started as one person or minstrel singing about what a good day they'd had and what a wonderful thing they had done (just like a man, to blow his own trumpet....) and as more joined in, they probably went and got instruments and joined in. There are incidents of several religious and not so religious processions using drums and rattles to scare the evil demons and spirits out of the woods before they went in.
If anyone wants a more comical look at the hunting of wrens, I'd recommend a parodic poem/song by Les Barker, sung by Martin Carthy, album unknown, but probably Oranges and Lemmings, called Milder and Molder. Tongue firmly in cheek for that.....