As has been said, all can be done online via the DVLA:
https://www.gov.uk/personalised-vehicle-registration-numbers/take-private-number-offCost is £80 to place the private registration on retention (as of March 2024).
The V778 is what you will receive in the post. This entitles you to hold the plate for up to 10-years, at no further cost, and not assign it to a vehicle. After 10-years, you have to either pay another retention fee or the plate goes back into the system and can be repurchased by someone else.
In your case, this isn't relevant because you plan to reassign it, once you have the new V5C in your name. As has been mentioned above, worth asking the dealer to supply you with a set of plates in your wife's registration as this will save you the hassle of having them made up.
