Going by that logic, nobody is truly British today. More than half the British population is migrant today whether it be first generation or the 10th.
And secondly, in Britain, everybody still maintains and is encouraged to celebrate their original roots. In America and Canada (to some extent), you are encouraged or rather pushed to be American/Canadian first and Indian/Pakistani/Asian/African after that.
And again, it totally depends on the kind of demographics of your circle, a lot of brown people or other third world immigrants in Britain, tend to socialise according to their race, they might interact with others from similar races, but they never go all out to integrate in the society. As an Indian, who has worked extensively with people from all geographies ranging from Americans to Far East Asians, and having lived in Britain for a brief time, I have never encountered a single incident where I felt that I am being looked upon differently. I do get a lot of vibes from similar Indians/pakistanis that just because we were born in similar cultures, we should be acquainted or hand out favours to each other or form a sort of cult, however, I do not entertain them cz I do not subscribe to such ideology. It all depends on the kind of people you deal with, if you engage with rowdy uneducated whites or the guys from the hood, as they call them in the US or 'brave and chavs' as they are called in the UK, you will definitely be treated differently, because they are uneducated and not used to seeing or interacting with people who look different to them. Its really not that different to being a typical North Indian guy in a South Indian village. You'd get the same weird different racialised treatment there too. And trust me, whites are the most accommodating people today, their white supremacy of yesteryears has now been turned to white guilt by the woke generation in the name of diversity and they go out of their way to make you feel welcome and comfortable in their community.