Just to clarify what you're after..? There's a difference between a Private Room and an Amenity/Side Room.
A Private Room will generally be part of a private ward and all your care will be private. By all accounts these cost huge amounts!
An Amenity Room on the other hand is a side room in an NHS ward. Your care is still provided by the NHS staff and you get the same food/etc everyone else gets, but you have a bit more privacy (as well as peace and quiet!).
I can't advise much about Private Rooms, but on the off chance it's an Amenity Room you're after then I can tell you what I know.
I'm up North as well, but I was recently advised that at LGI (Leeds General Infirmary) Amenity Rooms are either £45 without ensuite or £65 with ensuite.
However! There are only four Amenity Rooms available in total and they are given out according to need, i.e. mainly for patients who had C-sections or twins. They also cannot be prebooked.
Patients willing to pay can acquire a room if one is available, but in the event that the room is deemed to be needed for someone else, the paying patient can be turfed out at any point in the day or night and doesn't get their money back.
Not every hospital will be the same though, so you're probably best asking them directly. If you're having parent craft classes at the hospital, they'll tell you most of the info then. (Or that's a good time to ask them.)
HTHs!