Huge amounts of the NHS is already privatised.
GP surgeries are mostly private businesses owned by the GPs., As are dentists, opticians, etc., even those doing mostly/wholly NHS work.
Blair/Brown "privatised" many hospital buildings due to their love of PFI, so many buildings used by the NHS are privatised, as are the services within them such as catering, maintenance, even admin.
Lots of NHS services are contracted out to private firms, such as screening and many types of treatment.
It's been happening under BOTH Labour and Tory governments.
Funny thing is, that a lot of the private sections are actually more efficient!
Nothing wrong with privatisation if done properly. Trouble with PFI was that Blair/Brown didn't think about the long term consequences - they happily signed contracts spanning many decades without care of the overall costs as all they wanted were shiny new hospitals to show off to the media, with atriums, shops and cafes. Had they any sense, they'd have just built functional hospitals at a fraction of the cost.