There are a great many humorous snatches set to that tune, but they aren't necessarily related in any other way.

No real evidence that the first song is particularly "Irish", though it has been found there; as well as in England, Scotland and Canada. It appears in the Roud Folk Song Index at number 8248. Iona and Peter Opie (Oxford Dictionary of Nursery Rhymes no. 155, mention that it was sung to 'Brighton Camp' / 'The Girl I left Behind Me' in Warwickshire, and to 'Merrily Danced the Quaker's Wife' in Scotland.

Kenneth Peacock (Songs of the Newfoundland Outports, I, 55-56) prints a long text set to a very different tune. He certainly thought that the humour was Irish, but that may have been partly because the example he met mentions St Patrick's Day in the first verse. The tune is vaguely familiar, but at present I can't place it.

Paul's feeling that the song "mocks Irish poverty" is probably wrong (though there were certainly plenty of humorous 19th century songs on similar lines, many of which were written in cod rural dialects including "Irish"); it actually seems to be a parody of a London stage song of the early 19th century, 'Country fashions', which was set to the tune of '[the] Mon at Mester Grundy's', a slightly earlier song written in a form of Lancashire dialect (though as like as not also a London stage song). See  Bodleian Library Broadside Ballads:

Country fashions

Th' mon at M[este]r Grundys