I've no idea what happens at Heathrow now but it used to be that there were two buttons on the telephone keypad along with other buttons for other agencies. Controllers were aware of the emergency categories and would decide which was appropriate. The "red" button (if you like to call it that) was used when there was either an Aircraft Ground Incident (AGI) which could be, for example, a fire on an aircraft or something similar or an Aircraft Accident imminent or an Air Craft Accident. The initial call gave brief information on the location of the aircraft and the appropriate Rendezvous Point (decided by the controller). Within 3 minutes a further call was made with greater detail. The ther button on the keypad was the AFS Emergency Line which was used to notify a Full Emergency or a Local Standby.
Every thing was then taken care of by the emergency services and the aiurport PBX and ATC had no other involvement. It all worked like a charm.... but it's probably long changed.