We recently had a child in England ... a brief summary of the process is below
In England there is no prenatal care, there is antenatal care (which I thought when I first heard it that they were against babies). And you only see the doctor once in the process, the rest is handled by midwives. Socialised medicine is strange in what they do for you.
For instance, patients keep their own records, there doesn't appear to be a copy at the doctor's office (though this cannot really be the case it is much too absurd for a bureaucracy not to keep data), much less a digital version (and their project for that, the world's largest civilian IT program seems to be having difficulties and ... more difficulties (and its leader has come under some amusing scrutiny (his mom notes he failed computers in college) though they are still trying).
The fact that it is a centralized 10 year program already shows why it is doomed to fail, and that no lessons from the failure of transportation mega-projects and ITS or the success of the internet have been learned.
However, there are several home visits by the midwives, I suppose so the nanny state can check you out at home. They don't do ultrasounds either except when something appears to be a problem, those you must acquire privately. And for your blood work they give you Lucozade) rather than a special sugar drink (which I am sure is cheaper). Given the lack of doctors, lack of ultrasounds, lack of drugs, lack of malpractice insurance and lawsuits, lack of billing, kicking you out of the hospital in one day etc., it should be cheaper, one wonders why National Health Service are running a deficit.
The "A team" of doctors is at the hospitals (My wife actually only saw one doctor prior to the day of birth, and that was only to do the birthing plan, and he didn't even lay hands on). The procedure was only delayed 4 hours, not too bad on the whole. The nursing staff/midwives is largely immigrant, though I guess the US is moving that way as well. The midwives also seem to have less training than US midwives. The doctors seemed quite good (more competent than their US counterparts).
One doesn't get a private room. There is a ward with about 6 mothers, though there are privacy curtains. This seems to be more for the convenience of the nurses/midwives monitoring everything than to actually save money, the amount of space difference is minimal, and the hospital (Chelsea and Westminster) has plenty of enclosed open space. I don't know its utilization, it seemed higher than Fairview-Riverside.
Post-natal care involves 3 home visits by midwives, basically to collect data, again they don't lay hands on or even look closely at the baby unless asked to. They do weigh it once or twice. There is also a health visitor who comes by.
The at-home visits are nice in principle especially for late in pregnancy and just after child-birth when mobility might be constrained. In practice, there didn't seem to be too much point, the medical system just asking the same questions over and over again without actually treating anything. You get a team of midwives, so you may never see the same midwife twice.
The child-birth was much smoother this time, probably because a planned c-section is much preferrred to an emergency c-section. The lack of billing (or especially the infuriatingly time-wasting and tree-killing "this is not a bill" statements) is fantastic.