There was some case law about this a few years back but I cannot find it.
I am fairly sure the case involved a private hospital detaining someone who had gone in for a consultation and then charging them and I am sure it was held to be unlawful.
If someone goes into a private hospital voluntarily and is then detained surely there should be a mechanism by which the NHS take over the funding. It seems counterintuitive that you can be detained and charged for the period of detention.
Although it seems the main issue was the fact that the hospital detained the patient on s5(2) despite her only attending for an outpatient appointment.
I had a case in Scotland where my client was charged for her private care after she was sectioned . I took it to the Scottish Ombudsman and we received a full refund from Feb – August on her fees.