The UK spent most of the 20th century getting involved in other countries' politics. Especially in the Middle East. And Muslims have been emigrating to the UK for decades. Nothing new under the sun.
Not to mention most of those immigrating to Britain just so happened to be people from former British colonies, so they're just basically going to the country that used to own them.
Yep. The British marched in and told the locals "You now belong to Britain". Naturally lots of them showed up on Britain's doorstep. A by-product of colonization.
There's a reason why what first comes to mind when you say "Asian" in Britain is different from what comes to mind in the US, and why they have a lot of Jamaicans but not people from Martinique or the Dominican Republic.