I think it must be difficult growing up as the child of immigrants - trying to assimilate into a different culture to that of your parents. And so many of the Irish people who emigrated didn't do so out of choice, but out of economic necessity.
I've no direct experience of this, but my understanding is that many of the Irish who emigrated in the '50s, '60s, '70s etc. retained very close links to Ireland. They attended Catholic churches, they were members of Irish clubs, they drank in Irish pubs, their children often attended Catholic schools and were signed up to Irish dance classes or played Gaelic sports. They celebrated St. Patrick's Day. London was close enough to Ireland to allow for regular visits home. Their children were raised in the Irish culture, albeit in a different country.
I'm not sure what the experience is like for the children of Irish emigrants in London these days.
Anyway, I'm not in the business of challenging people about their nationality - whether that's someone whose ancestors emigrated from Ireland to the US in the 19th century or someone who emigrated to Ireland from another country 10 years ago.
I have an English friend who moved to Ireland as an adult. She said at this point she no longer feels fully English but also doesn't feel fully Irish.