As boundaries blur and the cryptocurrency sector gets more sophisticated, buying bitcoin in the UK is very similar to buying bitcoin in the US (see this article for a more detailed explanation of the process).
The market for bitcoin in British pounds is not one of the more liquid ones - according to CryptoCompare it usually ranks around the bottom of the top 10 most active markets in the world, well behind Korea, Vietnam, even Poland. However, UK residents wishing to actively trade or occasionally purchase some of the cryptocurrency have several reputable options to choose from.
Among the leading UK-registered exchanges are Coinfloor and CEX. (At time of writing, CEX has temporarily suspended new account registrations.) Both require user identification, and appear more geared towards businesses and active traders than the small retail investor.
Other non-UK exchanges such as Coinroom (based in Poland) and Kraken (based in the US) can also exchange British pounds for bitcoin with reasonable liquidity.
If you're more interested in a marketplace, BitBargain and Bittylicious will put you in touch with a seller in the UK (although only online or mobile payments are accepted).
And if you really want to buy bitcoin with cash, LocalBitcoins is your best bet, or you can try to find a bitcoin ATM near you.
(Note: specific businesses mentioned here are not the only options available, and should not be taken as a recommendation.)