Basically what I do is send the necessary forms to the DVLA to place my personal number on retention. The send me a Retention Certificate which I give to the dealer for him to register the new car with my personal number. Only once have I been asked to take the xisting car to a DVLA office for the details to be checked against the V5. I also receive a certificate from the DVLA giving a new number for my existing car and a form which allows me to purchase a new set of number plates with the new number they have allocated to my existing car. This is also given to the dealer and they make up a set a plates to go on to the car when it's handed over to them. I also receive a new V5 for my existing car some days later showing the new registration which goes with the car to the dealer. The new V5 for the new car with my personal number arrives in the post some days or a week or so after I've taken delivery.
This is the same process I have used for all nine transfers that I've done with my plate. I suppose in theory the new number should go onto my existing car as soon as I receive the details of the new number that has been allocated to it, but the DVLA do not know when those details arrive with me. It could be just the morning that the change of cars is taking place depending on the Royal Mail. My dealer has always said not to bother as they will make up the new plates and they will always certify to the DVLA, if necessary, that both cars were never on the road at the same time. They also say it is not a real problem as at the time of transfer and until I send them the part of the V5 signed by both myself and the dealer to say I've sold the car to them, I am the registered keeper of both cars.
In the past the dealer has been to the local DVLA office to arrange the transfer over the counter, but I understand the DVLA will no longer do this and the dealer does all the registration process for the new car by a computer link to the DVLA.