The problem there is, as already mentioned, there is british culture, and then there is british culture.
Everywhere in britain, there are different cultures to be found, go back 70 years or so, and you'd find what are now british stereotypes, the yorkshire miners, the manchester mill workers, the dockworkers of liverpool, newcastle and glasgow. Even so far as the upper class nature of the home counties, the oooaaarrrr of the west country, these were the identifiable traits of british. It wasn't just all about the tea, or drinking too much, or bland food and stiff upper lip. It was that and so much more, all rolled into one. We were proud of who we were, and what our ancestors had done.
Now however, with each passing decade, we become more and more dilute, homogenised and there is no real feeling of identity anymore. It's not simply immigrants, it is just the way that the world is, people move around more, and over time, that has a knock on effect.