Simply paying for it, and having that confirmed does NOT mean you can put the number plates on your car and start driving around.
Buying the private number plate is the first step. But, until it's been transferred to your vehicle, you cannot use it.
Your car has to be re-registered with the DVLA, under the new registration number.
That means getting all the documents changed for your car. You need to wait for that to happen, and be confirmed to you. When the transfer of your private number to your car is complete, you will receive the following as a minimum:
- Replacement tax disc
- Replacement mot (if applicable)
- Replacement V5 Registration Document (log book).
You'll get the first two items above as soon as the transfer is done. As soon as you have them, all you need to do is put the new tax disc in the window and the number plates on the car. You don't need to wait for the V5.
Don't forget your car insurance
You also need to inform your insurance company so they can amend your insurance policy schedule.
If you start driving with the new plates on...before the transfer is done and replacement documents issued... you are effectively committing an offence. So make sure you have the right paperwork confirmed first. Most companies will offer this service as part of purchase, or you can re-register the number plate yourself.