To avoid anyone having to plough through the article, I discuss in it, briefly, the tune commonly known as "Tell me ma when I get home".(Those words, like "love is but a lassie yet" define the polka rhythm). The children's song refers to Belfast City, and is now ubiquitous in Ireland, due I suspect to the Clanceys and other groups. Johnny O'Leary, the legendary Sliabh luchra accordionist, played it in the 50's and called it Din Tarrant's Polka, I believe (presumably because he didn't know, or had forgotten, its name). How long it had been around there, who knows?
William Irwin, in Cumbria in northwest England in 1860ish, called it the King's Polka. (Perhaps Johnny O'Leary, a good Republican, didn't like the title?). O'Leary and Irwin, incidentally, both played the same second part to the tune.
I don't know where you can hear an O'Leary recording,but you can hear the Lake District tune here(I hope). You'll need to click on the tune set called Dear Tobacco, and wait for the third tune. Don't be put off by the disorted fiddle sound on the second tune, I need to sort that out, it seems to have spontaneously degraded.
Anyway, I hope the relationship, geographical and chronol0gical, is of interest.