It's a matter of how you grow up. My dad's English (Canterbury) and my mother's Irish (Mayo), but being born in Ontario (which is in self-denial about just how English it is) and growing up there and Oxfordshire, I always felt more English. That said, I've got several hundred times more Irish relatives (how is that not surprising?), about half of which live in England. So, I guess I'm about as Irish as the Republic...
As for the real point of this thread, the answer is simple: get England Catholic. Say, some way, some how, random act of mass political stupidity, Bonnie Prince Charles wins England. He purges the government, and brings in a whole ton of Irish loyalists. With continuing Protestant threats, this evolves into a perpetual Irish governing class. Simultaneously, the Catholic Kings create a powerful Irish nobility, granting them estates in England and Scotland. (Basically the reverse of OTL) Eventually, the "British" Empire effectively becomes the Irish Empire...
Simon