I wouldn't be so worried. If you go into natural labour you don't get your regular doctor unless they are the one on call that day/night. It's not so bad, with my first I went through three different doctors just on delivery day. It really wasnt bad. They are all licensed and know what they're doing. Hope that eases your mind a little :)
Wow that's crazy they wouldn't tell you something so important geez! I had a NST today and all they do is wrap bands around your stomach and hook you up to a machine and monitor you and baby for 30 mins to be sure that the baby's heartbeat is normal and movements are normal. Also shows your contractions. Good luck with everything!