some one born in london, who has irish parents or irish grandparents and holds irish citizenship and irish nationality, therefore is irish born in london. there are also forms like liverpool, manchester and birmingham irish.
shane macgowan is a famous london irish person, lead singer of the irish band the pogues and noted as one of irelands national treasures, liam and noel gallagher are a good example of manchester irish as they both have irish nationality and were born in manchester.
English people with Irish heritage, basically exactly the same situation as Irish-American but sub out the American for English. English-Irish are sometimes looked upon as not real Irish because their families had to emmigrate.
London Irish or English-Irish are sometmes considered "not Irish" by the Irish born, just as Irish-Americans are looked upon as not "Irish"....except for Shane MacGowan of course, he's Irish even if he's really English.