Britain has had influence from across the water for hundreds of years and it hasn't diminished our culture so far. Look how many of our words have French roots or French words with English roots or all words with Latin roots and so on. Chips are Belgian, Tea is Chinese and the Queen descends from Germans. The blending of other cultures happened long before the EU and it's great that it did. Some of the most British things ever are things we took from elsewhere.
If it wasn't for our ability to take things from other culture and make it our own we wouldn't have fish and chips, tea or most of the stuff in the British Museum.
