Would you consider me English or not?

My surname is French I traced my fathers side and it was 12 generations ago that the French side of me came to Britain (but Normans did invade England which are from France) and my mothers side has Irish side 6 generations back (Irish/Celts invaded Britain though) I was born in England and all my family consider themselves English is that correct? Just how far back do you go before you say you are not English?