Well, some of these songs which people are claiming as English or Scottish could just as well be Irish - either by origin or usage ... Raggle Taggle Gypsies, for example, can't be accurately traced to just one place. It all depends, though, if you want songs you can say are peculiarly Irish. If you are just looking at "songs sung in Ireland" so well and good, but if you want peculiarly Irish songs, well ... here's a few more you might not want on your original list:

"Whisky on a Sunday" - English, composed (probably still in copyright).

"Home Boys Home" and "Botany Bay" are normally thought to originate from England and "A Parting Glass" from Scotland.