Having lived abroad in many places I would say the reliance on humour in every day life is uniquely British. Dry, self depreciating wit. Bill Bryson has written extensively about it and I think he is spot on. We keep our sense of humour no matter what. We banter brutally with loved ones and are icily polite to enemies.
I would also say we do traditional ceremony in a uniquely British way and we do it well. Everything from Trooping of the Colour, the Lord Mayor's Parade, the State Opening of Parliament, local 11/11 parades.
Our arts, literature, writing, theatre. We have such a rich tradition of those things. Some of the greatest international franchises are British: James Bond, Agatha Christie, Harry Potter, Lord of the Rings. London theatres are some of the best in the world and it’s rare to have the network of regional theatres and regional art galleries and arts funding that we enjoy. British actors are vastly over represented in Hollywood. Then the music: the Beatles, Rolling Stones, Queen etc.
I would actually say multiculturalism is a big part of our culture. We have a history of interaction with other people and nations. Many cultures were isolationist, we were not not. Food is a good illustration of that. Going out for a curry is a very British tradition.
Pub culture. Whole family to the pub for a Sunday Roast. Fish and chips. BBQs held outside in defiance of the weather. A pasty eaten on a freezing beach behind a wind break in a hat in May.
Football, talking about football, talking about not being one of those people who talks about football. Horse racing, whole families betting on the Grand National, hats at Ascot. Rugby and sending your tiny skinny boys out in the freezing mud and horizontal rain to be pummeled and many of them growing up dedicated to it for life. Cricket on the village green with a tea break that lots of the community bake for.
School uniforms and each generation finding a way to bend the rules and ruin the look the school envisaged. Few countries have school uniforms and if they do they tend to try and look smart in it!