Very soon after airborne out of Sig when the No2 gulped its oil and had to be shut down. Eng still going through the checks when we get a"rib 2 overheat" on the other side. Immediate actions include throttle back the engines on that side. So heavy and on one engine, we were going down...
Co-pilot puts out the mayday to Sig ATC, Eng runs the checks...
Fortunately, once the air supply leavers were shut, the problem went away and the engines could be throttled up, but for a few moments...
As a "War Sory", thats a very serious incident that you are describing: a heavy 4-jet ac powered only by one engine and "we were going down.." I feel that I must comment. Either your flight deck crew went against all their training or you are mistaken. I prefer to believe the latter.
Firstly, no issues about the oil system failure. If the indications required a shut down, so be it.
The immediate actions for a Rib 2 overheat do not include throttling back the engines on the affected side. In fact, the requirement is for them to be left untouched if possible (probably at high power, if "very soon after airborne"). Ultimately, at a later stage in the drill, one of the engines might have needed to be throttled back, but only when it would have been appropriate to the overriding need to fly the aircraft safely. In this case, throttling one engine all the way back to idle, while another one was shut down, would have been a serious mistake. A slight reduction in power (usually to approx 90% HPRPM) to extinguish the warning, then dump fuel to a safe 2 engine operating weight (20 minutes) and finally idle/shut down the engine, would have been the airmanship answer. If you were going down, it was because the P1 woud have recognised that he would have to descend due to the loss of the 1st engine and the probable reduction in power on another one. At no time were you in danger. The mayday call was probably due to the "SOP" that when overseas the crew will get the attention of the ATC controller and receive their immediately required revised clearances without any language difficulties.
I cannot believe that the flight deck crew acted in the manner you describe.