In a BBC debate, Thomas Cawston (then with the think tank Reform) states:
Competition is a "bogey" word in the NHS. Yet this hostility to
competition and private providers is a uniquely British obsession.
[...] The reason for competition is that [it] can drive real
improvements in care.
Cawston was an author on several reports for Reform. One from 2010 provides evidence of the "ability of competition to drive up health standards and productivity". For example:
The Office for National
Statistics’ analysis of productivity has shown that public sector productivity declined on average by 0.3 per
cent a year between 1997 and 2007. Private sector productivity, in contrast, rose by 2.3 per cent on average
each year. These statistics hide the relative performance of spending in different areas – and in this respect
health spending has performed particularly poorly. The NHS has lagged behind the rest of the public sector
on delivering value for money.
A 2014 report further makes this point about competition. Quoting from the executive summary:
This report presents ten case studies of successful health reform.
Each case study shows how a radical change in the delivery of
healthcare can result in improved quality and productivity.
The case studies show that successful health reform leads to:
Reduced costs through integration and competition
In Rhode Island, a private company, Beacon Health Strategies, has
created an integrated care pathway for mental healthcare services,
which were previously highly fragmented. In one year the cost of
mental healthcare hospitalisations for children was cut by 20 per
cent.
In Massachusetts any willing providers were invited to organise
community services for low income elderly patients. Patient-centred
care management replaced uncoordinated services reducing the number
of nursing home admissions by up to 42 per cent.
Reduced costs through standardisation of clinical practice
In the United States MinuteClinic has developed strict protocols for a
range of routine services. This has allowed the clinics to use nurses
instead of more expensive doctors, enabling them to provide
consultations 30 to 50 per cent cheaper than a visit to a GP.
The Indian specialist maternity hospital LifeSprings has focused on
providing a limited number of procedures and used standardised
clinical protocols. Doctors’ productivity is four times higher than
non-specialist providers and prices are up to 50 per cent lower than
of market rates.
Greater patient safety through service reconfiguration
In Finland, the Pirkanmaa region closed joint replacement departments
in five hospitals and concentrated care at one specialist hospital.
The new hospital delivered complication rates below 1 per cent
compared to an average of up to 12 per cent for general hospitals.
The NHS in London moved emergency stroke care from 34 general
hospitals to 8 specialist units with dedicated staff. London now has
the highest standards of stroke care of any major international city.
Greater patient safety through better data
Birmingham University Hospitals took the initiative to develop its own
IT infrastructure to track medical errors and provide decision
support to front line clinicians. Medication errors were cut by 66
per cent and contributed to a 17 per cent drop in 30-day mortality.
The Cleveland Clinic in Ohio has published its clinical outcomes and
data used by the hospital leaders to manage productivity, benchmark
clinicians and improve quality. The hospital is one of the highest
ranked in the United States for quality but costs are half those of
equivalent providers.
(In the spirit of the Ideological Turing Test, I've done my best to pretend there are credible arguments for NHS privatisation. But I don't think I passed, because there are no such arguments. As mentioned in the comments to the original question, this isn't a position that any sane politician in the UK would take. Crawson is now with a PR company, and represents the economic interests of private insurers and health care providers.)
EDIT: Here is a pretty extensive report by the Institute of Economic Affairs that argues for abolition of the NHS. It relies on a lot of international comparisons to make that case that lack of competition and private delivery are producing bad outcomes. It even claims that:
In terms of outcomes, quality and efficiency, social health insurance
systems [with delivery of care by the private sector] are consistently ahead of the NHS on almost every available
measure.