Just for your info, when both parties have accounts at the same registration company, e.g. GoDaddy, then it's a lot easier. That's because you can just do an internal account transfer, and once it is in buyers ac, they can manually change the WHOIS via the account.
So, one option might be for buyer to open an account at Namecheap, but they may not want to.
So, if not, they will just have to use the normal process, controlled by the 123-reg Transfer In option in their control panel. How slick this is depends on how well 123-reg has done it. At GoDaddy it's pretty good, you buy a Transfer In, and you step through the process. That puts the buyers ac in a position to receive the domain, and as the seller you just have to request the Transfer authorisation code and give it to the buyer.
Rgds