EDIT: Specifically regarding maternity wards: We did pay extra for a private room when my son was born, so I have first hand experience with that.
Mind you this was 13 years ago in the old maternity hospital here in Edinburgh - don't know what the arrangement is in the new Royal Infirmary.
NB My wife gave birth in a private room in the midwife run "Normal Delivery Unit" which was fantastic - we only asked for a private room a few hours after she had given birth after she had been moved to a ward.
Overall the NHS trusts have quite a bit of latitude in carrying out private services to offset costs by increasing utilisation of facilities and equipment (private surgeries are often carried out in NHS operating rooms, for example), but the extent to which they take advantage of it varies quite a bit.
I had my first child in a great hospital with the same room all to my family for our full two-day stay.
For my current pregnancy, I switched care providers, but I chose the new midwife based on her ability to deliver at the great hospital where I had my first child.
Last month (when I was seven months pregnant), my insurance company decided they no longer cover the awesome hospital. My midwife said insurance companies almost never let you stick with the hospital you'd planned in these situations. So unless I want to pay for the whole thing (despite being insured at great cost to my employer and me), I can spend my first days with my new baby sharing a room with some random family of strangers and their new baby, at a hospital where there's a high likelihood that the other family's new baby will be enduring significant medical problems.
Hooray for privatized health care, right? Clearly better than any other possibility.
That's very much all-or-nothing though.