A variety of options, varying at the state level
, enable illegal immigrants to receive Medicaid while pregnant. So plenty have Medicaid while pregnant and 4-8 weeks afterwards.
Yes I'm an immigration hawk delivering tons of "anchor babies".
To answer your overall question, we accept cash, give charity services, and setup payment plans.
But none of that figures out how to get them into a hospital. Hospitals only have to take them for emergency care, if they have insurance, or if they pay up front. This isn't a big deal usually for a cash paying illegal immigrant, as they will get care if they to deliver under EMTALA. But if I need to send them to the hospital for some non emergency test as a best practice for prenatal care I can't do that without insurance. That increases my liability significantly and as such, we do our best to not have any of those patients (we don't, they all go get Medicaid).