Note that "white British" is just the tallied report for English, Scottish and Welsh as well as generic "British."
"English" is more of a "white" identity. 72% of "white British" in England are actually "English" responses." Even among Jews - who are mostly third and fourth generation Britons of Eastern European ancestry - a majority say they're "English."
In contrast, South Asian and Black groups report mostly "British."
http://www.ethnicity.ac.uk/medialibr...ls-british.pdf
As for the Irish in London, yes, "English" or "British" seems to start with the second generation. Only 2% of Londoners are from the "white Irish" ethnic group which is synonymous more or less with Irish immigrants.
One thing to note about London too is that Irish immigration is more of a 20th century phenomenon than a 19th century one.
In Liverpool, only 1.4% are "white Irish." Obviously it is much higher than that - given the substantial Irish immigration it received in the mid-19th century. I'd guess at least a third are of Irish ancestry, I've heard as high as 50%.