Can I add one note? I hope it might help a tiny bit. Yes it can take a very long time to get in to the embassy to get the US passport and such, but it isn't a given that it will (as long as you have got your passport back, which you would if you withdrew the application, but that's part of the decision process you need to be looking at here).
I was able to get my daughter's consular report, passport, etc in two weeks from beginning to end. When I got on to make the appointment there was one for a few days from the date that I looked. I booked it. I had the passport in hand 1.5 weeks later. Sometimes appointments get cancelled and you can get right in if you time it right. I'm also wondering if there may be some way to push this for an emergency. Of course, it will become even more complicated for the OP if she doesn't have her own US passport because it has been withheld following a visa refusal.
Also, people aren't trying to yell at you. We're worried for you because you've taken some very bad advice, so we're trying to help you find a way to sort it. The reason it was phrased strongly was because you have come back saying you understand, but that you are going to try doing it a different way. It makes it sound as if, for instance, you aren't understanding that there is a risk that you could come back to the States with your baby, and without a US passport, she would not be allowed to enter the United States with you--then your life would feel an awful lot more destroyed.
I don't know that you would even make it that far. Nowadays they often check your visa status at the ticket desk, and if your daughter doesn't have the right status to enter the US, she might not be able to board the plane, and you would have to leave the country to remain legal. I would be beyond shocked if she got past the process with just a birth certificate because the US has really tightened up it's laws to protect parents from international kidnapping of their children, so there are strict rules about needing passports and needing to prove that both parents approved that passport. I've heard stories of people managing to get a child in with a British passport, but I think they've usually been travelling with both parents, so it didn't look odd to have two immediate relatives with completely different passports.
I agree with the others that a good solicitor is going to be essential, especially if you chose to remain here and not withdraw your application--you will almost certainly fail without a good immigration solicitor, even in appeal. I do suspect your application for a visa in the future (when you're old enough) will be worlds simpler if you withdraw and simply have the overstay and pay off your NHS bill. I know it isn't always that simple with expenses, having to leave family and such, but we've seen a lot of people get visas who overstayed without an understanding of things as long as they were totally honest, came back as fully legal spouses under the right visa conditions, and made any reparations for mistakes they made like getting benefits, paying medical bills, etc.
