If my car is worth over 5 times my debt and it also has private number plates, can they sell them together?

Yes

Bailiffs sell the car with the number plates it came with, even if the plates are worth more than the vehicle itself.^[1]

If a bailiff has taken a car with cherished number plates, the debtor can de-register the number plates leaving the controlled vehicle unregistered.

The bailiff only takes control of the car with physical number plates attached to it, the bailiff does not get the number plate registration certificate - unless the certificate is given to the bailiff, or it is left inside the controlled vehicle.

If a cherish plated vehicle is sold by bailiffs, the debtor can download and complete DVLA form V888^[2] and ask the buyer for the amount they paid for the car with the number plates and ask the bailiff company for a statement of account and see whether the bailiff was truthful about the genuine auction price the vehicle sold for.