I was born in London. My Dad was born in County Down in NI and both his Mum and Dad came from all over the island of Ireland for generations.
My Mum was an East Ender but her Dad could trace his origins back to Huguenots, that fled France for England in the early 1800's. Her Mum was a Devonian, again going back to at least the 1600's.
I think being "British" is a state of mind as much as a factual statement of ethnicity. I certainly think of myself British first and then English, but I can't say I am bothered overly much either way. There are more important things to be obsessed about !