To be a nit-picky econ type, the correct term for care in the U.K. is not "free", it is "paid by the government."
My insurance company negotiates a universal pregnancy care reimbursement rate with local hospitals. The universal reimbursement covers the standard package of ante-natal visits, a standard run of tests (GD testing, blood typing, dating and diagnostic ultrasounds), labor and delivery (including anesthesia if needed or desired), and post-partum visits. I don't know what this rate is (I suspect it's under $10K), I certainly do not have the leverage to negotiate it myself.
I do have some of the Explanation of Benefits statements from my insurance for my daughter's birth though. The full, billed cost of my c-section was something in the range of $40K, before the discounts applying to my particular insurance kicked in. The cost of a day of NICU care varied - her first few days were something like $20K each, once she got to be more of a feeder/grower, the cost fell to something more closely resembling $5K. She was in the NICU for 32 days. In every case, the billed rate was reduced to my health insurance company's negotiated rate, which was drastically lower.
If you find yourself in a position to give birth in a U.S. hospital without insurance, it would greatly help to be able to claim that you qualify for charity care, or to apply for Medicaid (which also has income limits). People really have wound up in bankruptcy over things like this.