From what I can gather, it depends on the hospital as to if they offer private rooms in maternity units and also the cost may vary.
Best bet is to check with your hospital and enquire as to availibility and costs etc and how to book, if possible.
One hospital trust I've just read up on said they don't book in advance, its a first come first serve basis as there is never any way to predict when someone will go into labour. Also the cost for that one is £100 per night.
Also found this online
6. Private room in an NHS hospital
Often referred to as an "amenity room", these are private rooms on the postnatal wards of NHS hospitals, although not all units have them. Midwives allocate them on the basis of clinical need eg if someone's had a caesarean, but you may be able to book one if there's availability after your baby's birth.
Pros:Having your own room means you can sleep when you want (or when your baby lets you!), see visitors in private and not have to worry about being "on view" all the time.
Cons: If another woman needs the room for medical reasons, she'll take priority. You may lose out on camaraderie with other new mums on the ward.
Cost: from around £45 a day, ask about availability when you go on your hospital tour.
Cheaper alternatives: as long as your birth is uncomplicated and you and baby are well, you can go home to familiar surroundings as soon as you like after the birth. Alternatively, there are still some small midwife-run units where the atmosphere is intimate and relaxed - ask your midwife about what's available.