I am not English. I work with English people, study with English, share a house with English. I don't know a single Polish person in Manchester (how sad is that lol). I am most definitely integrated but it does not make me English.
I have Polish passport and Polish citizenship not to mention Polish roots and family back home. I may one day become british citizen but evrn then I will have a dual nationality. I will always be more Polish then English simply becouse I spend my formative years in Poland and what I understand by my culture is Polish (sorry but no turkey for Christmas on my table).
You may think about yourself as English but I wonder how many people around you think about you as such.