Hi, once you have agreed a sale, providing it's a .UK domain, you would pay the seller, they then transfer the domain to you via their control panel at Nominet, that sends an email to you, click the link in the email and follow the instructions, the buyer usually pays the Nominet transfer fee (£10 + VAT) and it will take you through the payment process after clicking the link, you will need a credit or dedit card for that, Nominet does not accept PayPal.
As soon as you have paid the transfer fee, the domain is now yours and you will be issued with a Nominet account for that domain to administer it.
At the same time, you can set the TAG of your registrar during the transfer process, you can then transfer the domain in to whatever registrar you are using, on 123-reg.co.uk for example, it's free to transfer in .UK domains, so if you had an account at 123-reg, during the domain transfer process, you would set the registrars TAG as 123-REG and then transfer the domain to your account on there after completing the transfer.