Nothing has changed since I had my child at a London Hospital over 12 years ago. Back then they were pushing for woman to give birth at home with a mid wife. If there were any “problems” you could call an ambulance or drive to the hospital.
That may have been one hospital.
I worked for the NHS 14 yrs ago, and what you described was not general policy or what happened, then or now.
The NHS has problems, but it is far from the hellhole Americans think it is. People, and most Americans, go on what they read or hear from conservative media in the US.
Very few like you have any experience of it.