Mobile phone services are cheap in the UK, it is just a matter of finding the best value for money service that suits your needs.
Signal is pretty good in London although if you want to check the specific area you will be living in you may want to try Opensignal which provides an excellent guide on network coverage.
If you are new to the United Kingdom you may not be able to get a phone contract, however, there are some good pay as you go/pay monthly services that offer excellent value. Two provider's we recommend are The Peoples Operator (Runs on EE Network and 10% of what you pay goes to the charity of your choice) and Giff Gaff (Runs on O2 Network)
You can either go to the main phone networks EE, Vodafone, Three and O2 to source the deal best suited to you, or, alternatively there are quite a few resellers, here are a few major ones BT (EE network), Giff Gaff (O2 network), Sky (O2 network), Virgin Mobile (EE network), Tesco Mobile (O2), The Peoples Operator (Three network).
Relocating to the UK from another country or swapping phone service networks may mean you have to unlock your phone to use it. We recommend trying these three steps.
Step 1: Use Giff Gaff's Unlocking phone information
Step 2: If Giff's Gaff's unlocking service does not work get the Make and Model number of your phone, the network it is currently on and the IMEI number which you can usually find by entering #06# into your handset. Use google to see if you can find the information relevant to your phone to help you unlock it for free.
Step 3: Use an online mobile unlocking service provider such as Mobile Unlocked (this will cost you a little bit of money) or head to your high street and look for a business offering this service. Areas popular for travelers such as Earls Court, Bayswater, Kings Cross have quite a few businesses that offer mobile unlocking services.