There was an interesting programme I saw a good couple of years ago. It was a Scottish comedian, who just so happened to be a Sikh Indian (I think) as well. He travelled up and Britain to find material and people's opinions. He was a very happy Scot, it just so happened his parents were Indian and Sikh too. I don't think he could say he was 100% British (or Scottish... it doesn't really matter as most people would acknowledge they were British in the grand scheme of things), but I'm sure he felt a very strong allegiance to Scotland/Britain, but importantly to his heritage too.
I hope he's a good example of most Asians in this country, certainly the Asians I know (mostly Sikh or Hindu Indians) seem/act/feel as British as I feel, but they just have a different heritage that they must feel some allegiance too. Part of my 'feeling British' is being proud of my heritage, so surely 2nd or 3rd generation Asians feel strong and proud ties to India or Pakistan. I wouldn't expect anything else.
/edit: I'm on good terms with a lot of guys down the gym who are of Jamaican/West Indian descent. They would say they were British, mostly, but would say they were Jamaican first - even the 2nd generation guys. I think it's their kids who would simply say they were British, but had Jamaican heritage.
Also, a British-Indian guy down the gym was an interesting guy to talk to. He grew up in Manchester and would proudly tell you he's a Manc, and spoke of Polish/Eastern European's in the same way as I imagine many 1950's/60's Brits spoke of his family.