What fantasy Britain do you live in...
... where whole bureaucracies can be trimmed back or abolished, just because they're as useless as tits on a crocodile?
No, the *best* case is that there's a huge fuss about how hopelessly outdated the DVLA's database is, and a huge national campaign is launched to make everyone update their details and check them at least twice a year. Then those 7000 people can be gainfully employed with spamming everyone in the country to remind them.
A more likely scenario would be that the DVLA is made responsible for 'policing' car ownership, and you won't be able to buy or sell a car without logging onto their database, identifying both parties to the transaction, creating an ID for the guy you're selling to and checking that his Brazilian driving license does indeed allow him to drive it away, certifying on pain of perjury that you haven't "modified" the car at all, applying for a rebate on your prepaid road tax, entering the odometer reading and viewing a history of every MOT, every tax disc and every reported accident the car has ever been near.
Basically, anything - *anything* - to avoid laying off civil servants.