I have heard some m/ws will not object if you stay in the bath/pool, even if it isn't policy. I've often wondered what they'd do if you refused to get out anyway?

You need the water deep enough so the baby's head cannot get above the water until it's completely born and lifted out. That's right, isn't it Trav?
WRT the birth if you refuse to get out the bath, not unheard of for the plug to be pooled so you are in a cold hard bath with no water. It helps if you don't push vocally so you trick them into thinking the birth is a while away though.

In reference to the water depth: Yes that's right

The hospital I had the twins in had baths in every room but strict warnings of no births in them because they were not deep enough. However they had a lovely pool which was massive and loads of room for turning around. I loved it