I had my last Motobility car registered with my private plate before taking delivery. I had the number on a retention certificate and I gave that to the dealer so when I collected the car, it had my plates on it. When I rang Motobility to early terminate the car, I paid them the £80 retention fee and about an hour later, I got an email with a registration number on it. Took it to Halfords who switched the plates and a few weeks later, received a new retention certificate from DVLA.
BTW, when the DVLA issue a plate to replace a private plate, they do not use the original car registration, that’s dead and gone. The take an unreleased registration from the year the car was registered and use that.