Paying through payments is just one way. If there's an emergency, the bill simply goes up. You don't need to be paid up before you give birth or anything, it's just a way to make paying easier, and payments could continue after baby is born.
If you'd hidden your pregnancy or otherwise rocked up the hospital to give birth without warning, you'd be cared for, and either insurance or you would be billed after. If you're footing the bill, you could then pay in installments you could afford, or work something out. Lots of emergency medical bills get reduced/written off.
Also, some women give birth in birth centers which might be cheaper than a hospital birth if you don't have insurance, and some give birth at home with private midwives. That's a whole other thread, since midwifery isn't an area of study the same way it is in the US - babies are delivered by specialized doctors - and I don't believe there's a government standard for midwives, you have to trust their qualifications if you're hiring one (I think there's some sort of registry/qualifying board but I think it's still private organization).
In my state, you can be put on a 72 hour hold in a mental health ward if you're deemed at risk, like a suicide attempt. I think it gets more complicated after that.