DWP and HMRC , DVLA could be ' hived off' but as they don't make any profits I doubt anyone would want to buy and run those!
The work of DVLA, and many other "public" agencies, is already being carried out by private companies. Other examples are TV licensing, visa processing, offender management (prisons etc), many elements of social care...
For local government, many have some of their functions outsourced e.g. HR, call centres, various support bodies, traffic wardens, environment wardens are a few. For all these hived off services, you still need the same number of people to do the job, the same management infrastructure & the same equipment. In addition the company carrying out the role isn't doing it for the public good, they need to make their profits on top of the costs, thereby costing more. I did a brief study on this back in the '90's when privatisation of Government Services was all the rage, and in most cases the cost went up considerably, whereas it had been justified by notional reductions in costs.
Sadly cutting numbers of public servants has often been a visible way of "doing something", and I know of at least one Central Government Agency which was privatised that was a revenue earning agency.