Similar to what the others have said, at my hospital we have a special bereavement room with a couple of big sofas, a TV, ensuite shower etc. Even then, our normal labour rooms still have reclining armchairs in (but admittedly no pretty bed linen).
Delivery pack would be kept in the cupboard and wouldn't be opened till as far on as possible, as obviously everything in it is sterile. No point opening it early otherwise you'll just have to open another one. Gloves are in the cupboard again. Don't use gowns, only normal plastic aprons. Should be kept somewhere on the wall.
There would be absolutely NO fetal heart equipment anywhere near that room and if there was, you would be bollocked for it by the senior midwife. It's portable so no reason for it to be in the room.
As for entonox....she might not have wanted it. I've done a couple of induced IUDs and the women haven't even had time to get to the needing entonox stage as the babies have delivered very quickly. Even then, she might have had pethidine or diamorphine or something, you don't know. At my hospital entonox is kept in portable cylinders outside the rooms so if she asked for it someone could have just wheeled it in for her. I agree there should have been hand gel, they probably thought people wouldn't notice.
I can't comment on the duvet thing as I didn't watch the episode but it might have been very quick and she was caught out. Ultimately it doesn't matter if she's on the duvet or not as it's going to go in the laundry anyway.
We use a moses basket for IUDs but I think some hospitals might use cribs. Am not entirely sure though.