I'm not living in UK, but I've been there for a couple of weeks as tourist and spoke to many immigrants from all over England and Wales during that time.
Well, I think most of Poles still call themselves Poles. Most of them came for some period of time, like 1, 2, 5 years. Some of them for better job or studies, some of them were just looking for some new adventures. Also, I think many of them will stay and become British one day. It's really easy to fall in love in Great Britain.. ;)
You ask about culture aspect.. I asked about it once. I think most of them don't know British culture at all. Actually it's hard to find it on the streets. Most of them are busy people who work hard, shopping, going to the pub or cinema at evnings and don't really know a lot about British folk culture.
Comparing to Ireland or Germany, I think there's lack of integration programmes / festivals for newcomers in United Kingdom.. But that's my private opinion.
If anyone has any advise for how i can carry out my research, it would be much appreciated!
I think that the best way to find out the answer for your question is ask your Polish friends, then you will get the answers from the first hand.