Private hospitals vary in what they can and do provide. I worked for one that had HDU and ICU so could do things like bypass surgery.
Although if you had problems after you would be seen by an OBGYN until your consultant could get in (as the hospital had maternity services it was the most logical RMO to have).
The OBGYNs were on a rota from an NHS hospital so they were paid by the NHS but the private hospital paid the NHS hospital for providing the service.
The hospital I worked at implanted pacemakers, but the other private hospital near by didn't so their patients came to us for implants and then we walked over to them to do final checks before discharge.
As a PP stated, you have to be careful who you send to a PH because of the facilities and the staffing.
Mostly NHS patients come along with some NHS staff.
@Crumbleandcake don't worry your choices are still there, the plebs from the NHS are basically renting a space and some other services.
You are safe to choose the hospital you like, although if you are having a pacemaker fitted then the Portland is probably not the best choice. Likewise John and Lizzies, probably not the best choice for maternity care.