When you take the old car in to the dealer you can put your private plate on retention, on line, at the dealership.
You must then wait for the (old car) V5 to be posted out to you, usually a couple of days, you then have to get it to the dealer asap so they can sell the car as they now own it.
You will get the V5 for your new car through in a couple of days, as well as, separately, the retention document for the private plate.
When you have both V5 and retention form you can transfer the private plate on to the new car on line.
The dealer may want to charge you a refundable deposit to 'motivate' you return the V5 to them asap.....