It's more complex than that really. Irishness is only one small part of Britain's multi-cultural & multi-ethnic make-up. It doesn't end in Ireland either, as it permeates British culture, and not just on the western coast of England (Lancashire) where it is admittedly prevelant and cherished, but by no means exclusively so. I myself have East Coast Tyneside Irish roots, but my only glimpse of the Emerald Isle was caught on a childhood holiday to the Isle of Man. Celtic culture on the British mainland is informed by Scotland, Wales and Cornwall but to what extent it exists above the romantic mists and nominal bi-lingual road-signs that mark the borders is difficult to say really. Experience tells me things are more solidly Celtic in Wales, but that doesn't mean the kids are playing crwths and pibgorns, which are, in any case, pretty recent reinventions. Even as recently as 20 years ago punters would ask me what that funny instrument was that I was playing; these days they're likely to tell you: ah! a crwth! (though Folkies still persist that anything played with a bow that isn't a fiddle must be a bowed psaltery, bless 'em!)

In the end it's all down to how people think of themselves as individuals rather than in terms of culture, folk-spurious or otherwise. I think it was Hamish Henderson who pointed out that before one can be truly National, one must first be International. I'd say that applies to being British too - before I can celebrate my Irish / Scottish / Northumbrian / Saxon heritage, I must first acknowledge that I share my soil with a multiplicity of ethnicities and cultures; that Britishness is also Asian, Chinese, Carribean, Jewish, Roma, Polish etc. There are no absolutes, just individuals, which maybe accounts for the sort of pedantry one finds in Nationalists, Folkies and Pagans, who are invariably people inclined to cultural correctness, fearful of the more feral & fluid nature of Actual British Culture both high and low which really couldn't care about identity above and beyond the normalcy of everyday life. In real terms, there is little room for The Celtic Twilight save dressing up as a Leprichaun on Saint Patrick's day and getting bladdered on Beamish stout in an Irish theme pub in Manchester whilst bawling along to Danny Boy.

All my Scottish friends live in England; the only friends I have in Scotland are English. Go figure!