I would argue that the National Health Service (NHS) should not be privatized. The main reason for this is that doing so would undermine the very principles on which Britain's health care system is based.
Under the NHS, health care is provided free of charge on the basis of clinical need. If the system became privatized the whole dynamic of care based on clinical need would change. Health care would be provided in order to make a profit for private companies. Inevitably, this would mean large numbers of patients being unable to access healthcare as they wouldn't be able to afford it.
There would also be the question of accountability. Most healthcare companies are listed on the Stock Exchange, which means that they have to answer to shareholders rather than patients. Shareholders naturally expect the companies in which they invest to turn a profit. But in a fundamental sense, this is at odds with the clinical needs of patients.
Of course, privatization can take many different forms, and in recent years private companies have taken on an increased range of services and functions within the NHS. Private sector involvement has been part of the NHS ever since its inception. At no point has the NHS ever manufactured its own surgical instruments or medicines. But if the entire system were privatized, as suggested in the original argument, then that would alter the whole ethos of the public health care system to such an extent that the NHS would no longer be the NHS.