You have to put the plate on retention, which cost £105 last time i did it, they keep your docs in the office and send out new mot and tax disc normally with 10-14 days depending on dvl a office then new reg doc can take up too 4weeks to come in.
When the plate is on retention it includes transfer fee so there is no charge to apply to new vehicle.
Putting a private number onto a car is normally done on the day but retention takes a little longer. Length of time depends mainly on your local office.