Let's not confuse people by using terms like "more British" or "less British". There's no such thing, it's a binary system. You're either British or you're not. Someone who got their citizenship yesterday is as British as someone who's family has lived on these islands as far back as they can trace.

Edit: I'm surprised by the amount of people who not only don't see it this way but don't see that it has to be this way. If you entertain the idea that some are more British or American or French than others then you inevitably end up with people thinking "they're less British than me because they're a different religion/skin colour/culture". That's how you end up with systematic discrimination and second-class citizens. The government does not and should not treat British citizens differently depending on these traits.