The DVLA will send you a V778 that you will enter on the DVLA assign a cherished plate site. This is done using details from the V5c and the V778, as soon as that is completed a new V5c is posted to the registered keeper within 5-10 working days.
As soon as the registration is assigned you can fit plates to the car, as is explained on the DVLA site. Use the V778 to have plates made up, the DVLA will not send physical plates to you. According to a guy I spoke with at the DVLA there is a 7 day grace period to fit the new plates, although I always have them ready to fit as soon as the transfer has been done.
The whole process can be done by post, but it takes so long that it just seems like a waste of time. Either way, both done by post or online it’s a relatively simple process. Online makes it easier as if you make a mistake such as a typo you know about it immediately.
remember before you sell, if you want to take the cherished plate off the car, it is £80, if it is not done before transferring ownership the cherished plate ownership goes to the new registered keeper.