The Economist looks at a survey conducted by Alan Manning and Sanchari Roy of the London School of Economics where they reworked data from the Labour Force Survey to show that immigrants from "problem" countries are more likely to describe themselves as British than those from Western European and North American countries. Partly this is because they may have to take on British citizenship in order to remain in the UK. But a more profound point brought out by YouGov interviews is that most ethnic minorities see Britishness as identified by values (fair play, rule of law etc) than by the icons that white Britons choose to define Britishness (fish and chips, Winston Churchill etc).
I have noticed a tendency for white Britons, particularly from the intellectual classes, to reject defining Britishness by values, because they say that these values are not particular to the UK. If you come from a mature democracy, this may be true, but those immigrants who don't, have experienced the fact that such values are not universal. Actually, this can be the case even with immigrants from mature democracy - many years ago I had a German neighbour when I lived in the East End of London who said he felt the British were much more tolerant of diversity than the Germans.