The last time official Government witnesses were present in the room for a British royal birth was on Christmas Day in 1936 (Princess Alexandra). King George VI abolished the requirement of Government witnesses shortly before the birth of Prince Charles in 1948, deeming the practice outdated and unnecessary.

Oh yeah, as an aside, we ran into that in L&D almost all the time. The worst was when an entire family showed up with food and planned on having dinner while watching the birth...they were escorted by security out the door.
Cynthia's answer about the officials in another room during the labor and delivery process quelled my "ick" reaction all these years. It makes much more sense now.