I'm an EU citizen as well, and it took me a bit of time to get used to British humour, but now I love it!
I'd advise you to watch Monty Python, as has been recommended, as well as Mr. Bean, Blackadder, and programs like Have I Got News For You.
A lot of British humour is delivered in a dry manner. So, watch the facial expressions of those who are delivering the joke, and the body language, especially with people like Rowan Atkinson.
Brits laugh at themselves and laugh at others. They laugh at serious situations, especially when things are bad. It's all about not taking yourself, or life, too seriously. They'll joke about politics, the state of the country, small mess ups and great, big, whopping ones. I like it, because when things look grim, it helps to laugh. When you laugh, it helps you to carry on.
British humour can be really subtle. Don't expect neon lights with flashing arrows pointing down stating "This Is A Joke!" It helps to have an understanding of British society, so that you know what the comedian is on about. I still don't know everything about it, so sometimes I haven't got a clue either and things go way over my head, but I'm learning
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Warning, though - once you've learned to understand, and hopefully love, British humour, other types of comedy can seem very over the top! I've watched movies and thought "yeah, yeah, I get it, I don't need it spelled out for me!" But that's a price worth paying.