Three UK is currently pushing out a carrier update to iPhone users that allows them to enable 4G LTE on an iPhone 5 or iPhone 5s. Users should be prompted to update their carrier settings at some point, though you may find that you need to go to Settings > About in order to receive the prompt.
The carrier software version should go from 15.5 to 15.6. After updating the settings users should be able to turn on LTE by flipping on the Enable 4G switch in Cellular settings.
Have you received this update yet, and if so, what do you think of Three's LTE? Let us know in the comments.