If you have a phone with someone other than Verizon and it's unlocked, you should be able to just switch to a SIM card only plan with one of the UK carriers. If you have a smartphone, you can always still use a US phone just on wifi in the UK to make calls and send messages via apps like Skype, iMessage, FaceTime, whatsapp, viber, etc. if your US phone is verizon or it isn't unlocked, then you'll probably have to buy a phone over there and get set up. Depending on how long you're staying, you may find it more cost effective to but a phone outright over there and use a pay as you go service rather than a contract. If you want to nose around and find out about UK carriers, the main ones are Vodafone, o2, everything everywhere (EE), 3 mobile, and tmobile. Someone else may have more advice for you too. But that's a start.
