The Good, the Bad, and the Crispy - S2-E18
Continuity mistake: When Lucifer is sending his mum to her new world, he is standing for the whole scene. They show a shot of Charlotte collapsed on the beach, and in the same shot, you see Lucifer on his knees in the sand. The very next shot, he is standing again. You only ever see him fallen on his knees when time starts back up again and the detectives realise where they are, which is a bit later. (00:37:25)
Answer: At least as far as clothing goes, I was in regular clothes when my wife gave birth, only changing into scrubs when they thought we'd have to move to an operating room (but ended up not needing to), otherwise I'd just have stayed in them. The room was also pretty much a regular hospital room, but no doubt there were some birth-specific differences I wasn't aware of!