NATS don't establish any form of airspace for Royal Flights.
This is done by the Airspace Utilisation Section (a joint civil/MoD organisation but nothing to do with NATS) based on the requirement given to them by 32 (The Royal) Squadron (who are also nothing to do with NATS
). As already mentioned, if existing Controlled Airspace is already there, then this is used. If it is not Class A however, it will be upgraded to Class A for the periods stated in the NOTAM. This is to prevent VFR flight within the airspace.
NATS comes in to play, usually as the Controlling Authority if they have responsibility in the airspace concerned, or to advise AUS if the airspace designated has errors in it on receipt of the NOTAM.