Pointing the new domain name at the site is easy, and something you will be able to do via the 123-Reg control panel.
The other thing that needs doing is getting wherever hosts the site to recognise that when someone asks for newname.co.uk, they want the site that's there already. That's going to be a couple of minutes work for someone, but who that is and how easy it will be to get them to do it depends on where it's hosted.
Oh, and presumably some of the site text may need changing.