Yes. Most car rental suppliers will permit you to add an additional driver to your booking. However, an extra fee may apply for any additional drivers. They will need to present their licence at the counter in order to be added to the rental agreement. Additional drivers can register at any branch they can't be there physically at the time of pick up and it can be done either before or after picking up the car rental.
Remember that few agencies consider the renter's immediate family as authorized drivers. Note also that additional drivers must meet all rental requirements and be present to sign the rental agreement.
VroomVroomVroom compiled a quick guide to the terms and conditions of each of our suppliers in the UK.
Read more about picking up a rental car in the UK.