Sarahhappyjoy's comment strikes me as particularly appropriate. If you're only a few weeks pregnant and otherwise healthy she's right, you don't actually NEED any services from the NHS.
One of the things I loved about the UK was that pregnancy was treated as something 'normal' rather than the 'condition' it seems to be in the US.
And ironically, the care turns out to be much better in so many areas of maternity care here with the NHS. I'm not trying to argue with the OP. I remember well how I felt a bit bereft with my first pregnancy as I missed out on some of the stuff that turned out to be cosmetic (like a fancy gyno/obgyn office and huge amazing maternity suites at new hospitals) or when things were just different (I'm still not wild about having to choose midwife-led deliveries or consultant-led and your experience being quite different if you want more medical stuff like an epidural. One thing that helped me was my sister in the US was pregnant at the same time as me, and I actually found that we met with our professionals at about the same intervals and had very similar levels of care. In the end, she was often telling me she was jealous of things I had on the NHS that she couldn't get in the US. The main difference being how it wasn't so medicalized here and I had to make more choices on my birth plan which dictated where I delivered more.
If you want to do research into maternity care though, take a look at infant mortality rates. Even with all it's fancy care, a lot more babies die in the USA than in the UK, and they certainly don't send people out to your house in the US for the first several weeks to do baby checks, breastfeeding support and weigh-ins instead of making you take a vulnerable baby to the doctors office full of sick people (when you yourself are still recovering and exhausted besides). There is so much that is actually better in the maternity care here. It just takes getting used to a different idea of what to expect.
Having said that, it doesn't sound as if you plan to be here to have the baby. If I were you, I'd be checking into Medicaid options for wherever you will eventually have the baby. You'll want to get that set up in the US so you'll be covered there (assuming you don't have insurance after being away for some time). Happily, it is generally much easier to get when you are pregnant--although it's difficult if you aren't.
And while I agree with what many have said here about fairness and paying into the system, in the interest of fairness, it must be said that the OP was right in one account. You actually don't necessarily have to pay for maternity care when you come from other countries. It isn't well publicized, but most states do have special Medicaid provisions for illegal entrants. If you came as a legal visitor, I'm sure you would have to pay for things, but many people from other countries do receive financial assistance--even in conservative states that hold on to Medicaid funds with very tight fists. It is usually done as a measure of not punishing innocent babies or putting their lives at risk because their parents weren't able to get insurance coverage. That Medicaid would go away as soon as baby is delivered, but there are many means of helping some migrant families to have safe deliveries.
Having said all of that, I am pretty sure that if the OP's role was reversed and she got pregnant in the US while on a regular visit visa, I'm sure she would have to pay for all her antenatal care there too.
Best of luck in the visa appeal and in finding people to provide care. I hope things work out for you. It must be a frightening time (and if you're anything like me, I'm guessing pregnancy hormones don't help with that stress level.) I'm near positive though that you aren't going to find coverage outside of paying for it in full. Even in the US, I don't know of any insurance companies that cover pre-existing pregnancy, and as others stated, insurance is totally different here and works in conjunction with the NHS anyway. So it will likely be a matter of just coughing up the funds or getting quickly back to the States and getting Medicaid or insurance.