Before the buyer disappears into the sunset with your old car, you need to do a couple of things.
First, write out two receipts with your details and the buyer's, along with the car's registration, VIN and description. Don't forget to include the price of the car and the vital phrase 'sold as seen'.
This doesn't mean you can lie about the car's condition, but, if you've been honest, it stops the buyer coming back later for a refund because the car's broken down. You and the buyer should sign each receipt.
The last, but most important, thing to do is fill out the V5C registration document. The buyer needs to complete their details, and take the new-keeper section, which helps show they now own the car.
You should take responsibility for posting the V5C to the DVLA. Unless a new V5C is issued, you'll be the first port of call for any speeding and parking fines, along with driving offences the new owner commits.
Failing to notify the DVLA about a change of keeper is an offence, and could get you into trouble if your car is involved in an accident.