To transfer a UK domain name - and specifically a co.uk domain name - you will need to do two things (as of 2014):
When running a whois search for a domain name address, you will notice that a domain name has a registrant and a registrar. The registrant should be in the owners name, and the registrar will be the company the owner paid to register the domain name on their behalf.
To change the registrar of a domain name - so the owner can manage and renew the domain name elsewhere - the owner will have to find a new company to host the domain name with, and, find out what their IPS Tag is.
Some registrar companies - like 123 reg - allow full DNS management, and within their account management panel domain name owners can change the IPS Tag of their domain name themselves.
The usual scenario, however, is that an owner of a domain name does not have full DNS management. In this scenario the owner of the domain name will have to contact their current registrar and ask them to change the IPS Tag for them. A good registrar will do this for free, but some may levy a charge.
A problem will occur if a registrar refuses to change the IPS Tag or does not respond to the request. In this situation the owner of the domain name will have to contact Nominet and ask them to change the IPS Tag; Nominet are currently (2014) charging £12 for this service.
A serious problem may occur if the registrar of the domain name has registered the registrant in their own name and not the "real" owners name. If the registrar is unwilling to change the IPS Tag in this situation, then it will become a "sticky" issue to get Nominet to change the IPS Tag.
Once a domain name has had it's IPS Tag changed by the current/old registrar, and the new registrar is aware of this (the transfer is pending in their system): then the new registrar will automatically be checking Nominet's database and attempting to complete the transfer. The process should be completed within an hour once the IPS Tag has been change by the old registrar.