Physical Danger
Unless the new mother is Daenerys Targaryen, the Mother of Dragons, you probably don't have a whole lot to worry about. During the birth there may be flailing limbs, but generally the mother and doctor will be the ones in the middle of the carnage.
You're never really in danger, but some of your patients might be feeling a little, shall we say, overwhelmed by the bowling ball trying to leave their body. While you take care of their needs, get good at ducking to avoid painkillers thrown at your head, and practice your one-two hop-skip so you can get down crowded hospital corridors without overturning any gurneys.
With some quick reflexes, you'll be fine. And with a good head on your shoulders, your patients will be fine too.