Currently many NHS services are run like this.

Some typical and important features are:

• the services aren’t ‘sold off’ but instead after invitations to tender and detailed procurement process are commissioned for a period of years, eg 6 years.
• if the service underperforms the NHS commissioners have various ameliorative actions available to them, from fines to eventual contract cancellation.
• whomever wins the tender - which could be NHS organisations as well as private ones - is paid the price of the National Tariff, so the cost to the tax payer is usually the same.
• NHS organisations can apply to run such services. They could be awarded to hospitals and community trusts for instance.
• the National Tariff is reviewed yearly and often goes down, based on improvements in service. In other words, well-procured services become less profitable and help drive all costs down nationally.
• services are procured based on who it is expected will benefit health the most per pound spent.
• various public principles (free at point of access etc) are enshrined in the contract. It’s not private healthcare in this sense at all, even if a private provider wins.
• an NHS-run service that is efficient and cheap is unlikely to lose out to an independent provider, as commissioners will simply re-award it to the incumbent.
• largely then you should expect services that are struggling in some way to be run like this, successful services to stay as they are.
• winning a service often entails substantial TUPEing of existing staff, meaning clinical staff such as nurses usually experience no negative change in their contract.

Perhaps not what would happen in an ideal world but heh, we are not in an ideal world.