Sorry long post. Your fault for asking a long question!
The NHS is already privatised in many ways. Think about things that exist in a hospital:
Doctors, nurses, HCAs are paid by the state. However, the chronic underfunding means we often have to rely on agency staff, who are working for private companies. Sometimes there are permanent rota spots given to private agencies. Porters and catering staff are often rented from private companies. Pharmacies are often run by Boots or similar particularly when compliance aids are involved. Older hospital buildings are generally owned by the hospital, but almost all new-builds are PFI i.e. rented from a private company. Sometimes hospitals don't even own their own CT or MRI scanners - also rented. Radiology services particularly out of hours are often out-sourced to e.g. Australia, India. Lots of secretarial work such as writing up clinic letters is also done in India. A lot of hospitals generally have at least one private ward patients have to pay to be on. Some services such as screening services are being offered to private corporations like Virgin and the running of one hospital was even given to Circle a few years ago, who first celebrated how easy running a hospital was
before promptly collapsing and having to be bailed out by the NHS at huge expense
A lot of this is to do with making short-term savings without much thought for long-term outlook. PFI in particular can be a crushing fixed cost placed on trusts who are now being told to make 10-15% "efficiency savings" but the 20-30% of their budget is ring-fenced so the remaining services are in fact looking at a 25-30% cut. That's huge.
We are looking at more privatisation in the future, Jeremy Hunt famously writing a book on the topic. As above, it could take many forms. The concern is that it will a) restrict free access to healthcare e.g. fees to see a GP and b) contracts will be given to JH's pals without fair assessment (lets not pretend corruption is not rife within the kind of corporations we're talking about) and c) you'll get the spiralling costs you get with private healthcare: at the moment doctors have at least some sense that they should not waste NHS resources, not request unnecessary scans or blood tests etc. But if its a private company paying and your neck is on the line (being sued is more likely if the patient is directly paying for their care) then they might just request all the tests to be safe. That's what happens in America, who do more than 6x more CT scans per head than us.
The locum agency pay rates were much higher than NHS pay rates, suggesting that a private system would result in much higher doctor's pay. But it depends on the system they implement - whilst the government has introduced all the private stuff above in the name of the free market, for doctor's they removed the free market by introducing a pay cap. Its free market economics but only if you're a mate of the Health Secretary, some sceptics might say.