This was my experience... perhaps I lucked out (well, I know I did!).
I was supposed to deliver my 1st at the HCMC midwife unit; he came too fast and we had to go to Mercy instead. I showed up pushing and they knew nothing about me.
My triage nurse was in CNM school; she stayed with me for the birth (1st stroke of luck).
The OB that I got happened to deliver my doula's kids... Dr Collins (2nd stroke of luck). He wanted to put an external monitor on me. I said no thanks. He said OK. He wanted to put in a heplock. I said no thanks. He said OK. He let the triage nurse/ CNM student handle most of the birth along with my DH and just sorta sat back out of the picture. He scooted forward for the crowning, checked for a cord, and let my DH and I pull him out the rest of the way. He asked if he could give me a pitocin shot after the birth. I said no thanks. He said OK. He asked if he could stitch up my "skidmark". I said no thanks. He said OK.
I don't think the hands off approach was his natural inclination but he was great!!
The night nurse did not object when I wanted to do his 1st bath. We were never separated.
I left after 24 hours. I think normally that's something that is discussed in the office so the staff knows and no one has to scramble to get their stuff done. But they were OK with it (well, the day nurse grumbled a bit but she was kinda an old grump).
It could've gone so much worse. But I don't think it could've gone better!! I think natural births don't happen there very often; my DH said people were talking about it at the desk.
Also I think people respected our wishes b/c we are both RNs and my DH works there. And I'm a NICU RN so I didn't need any help from the baby care/ BFing end of it. In fact the nursing staff just kinda left me alone, which was what I wanted
So there were a lot of stars in alignment there, but it can happen! I think in a hospital not set up for that sort of thing, you need to have your wishes known and have good guardians around you (ie doula and DH) to make sure the "norm" doesn't happen to you (EFM, heplock, epis, pit, etc).