The NHS to me is very different in the way pregnancy is dealt with. I have had two children in the US and am expecting my third. There is no private insurance that pays for a normal pregnancy, I have fully investigated this possibility. You can go private, in which you pay for treatment yourself. I have checked into the prices ranges from 8,000 to 15,000 pounds.
The NHS is not free, you do not have to pay for anything at point of service. My husband pays a large amount of his check monthly to cover his NHS contribution. I would perfer he paid for private insurance. I do think the NHS is an excellent thing to have, due to the fact that everyone is fortunate enough to get medical care, something the US lacks.
I think you will be more than fine to have a baby on the NHS. there are some extra tests that you can pay for yourself, if you choose.
I wish you the best of luck.