Lots of integrated people exist in the UK at present, but the whole point about integrating is to become the same as the rest. If you give out a message that says 'look at me, I'm all integrated' it removes your cloak of invisibility. So really, we don't know empirically how many people of foreign extraction live in the UK but have integrated.
That isn't to say that no problem exists regarding resisting integration. Certainly, in my little part of the country, there are numerous mainly Pakistani and Bangladeshi communities where integration is either consciously resisted or doesn't happen because of social practices.
I have noticed a shift over the 30+ years I've lived here, mainly with a huge rise in veiled women, and Muslim women who don't wear veils will (if they trust you as a friend) often relate how they are harrassed to distraction by men from their communities. This makes it harder to integrate, as being human you'd probably slap a scarf on just to get some peace.
Marrying someone from the subcontinent prolongs the existence of a non-English speaking mother or father, and that delays the integration of English as a first language into families.
Sadly - and I'll get shot down for saying it - in these communities in this area, many families don't see education as a priority, and at the same time the steady increase in extreme religious views has seen a retreat into opinions and learning being guided at mosque / madrassa level.
But taking a longer view.....Irish Catholics were frowned on for 'marrying out', for socialising beyond the parish community, or for questioning the parish priest's views. that's all changed now, so in the longer term, maybe the present difficulty will fade.