My thought is that this is something Labour has accused the Conservatives of constantly for decades and yet somehow it hasn't been privatised yet. It is on a par with their "24 hours to save the NHS" calls during election campaigns. The Conservatives have been in power for over half the time the NHS has existed and yet, somehow, they haven't destroyed it yet despite constant claims by Labour that they are going to.
There have always been areas of healthcare that are delivered by the private sector - dentistry, opticians and chemists, for example. I don't agree with those who insist that the NHS shouldn't use the private sector for anything.
For me, I don't have any ideological objection to services being provided by businesses. What matters is that the service remains free at the point of use, the cost to the taxpayer is no more than it would be with a public sector provider and the quality of care is at least as good as with a public sector provider. If those goals are met I don't personally care whether the service is provided by the public sector, the private sector or the third sector.