You'd need to enquire with the hospital trust. I've seen on netmums some hospitals charge anywhere upto £150 a night for a private room. As I understand it you cant prebook, you have to ask on arrival to Labour & Delivery, if they have one free. If your willing to pay then it's yours. They do prioritise though for ladies in need, such as emergency c-section and further complications or for ladies/babies with infections or for ladies with babies in NICU. So not sure whether you would be moved a lady in need came in. I'd definitely just check your trust's website or give them a ring.
In the hospital I use you have luck of the dice sometimes you can just be taken into one of the private rooms..but otherwise I know there is a £25 charge