His father C. Day-Lewis was the Poet Laureate not of "England" but of the United Kingdom, of which Northern Ireland is a part. C. Day-Lewis came from an old Protestant Ascendancy family in Ireland.
Daniel Day-Lewis identifies as Irish.
Listen to the few Daniel Day Lewis interviews. His accent is English. Day Lewis was brought up in England. His father is Anglo Irish. This is very different from being Northern Irish. His paternal line is from Anglo Irish aristocracy in Southern Ireland, not Ulster protestants. He claimed sole British nationality in 1948.