In this thread:
AC maydays not accommodated at Hyderbad until the 4th call there is discussion of 'mayday' calls and what happens then.
If a plane has declared 'Mayday' and needs to land as soon as the suitable runway is reached, but the ATC for whatever reason isn't giving priority, I gather the plane, in distress and if necessary, will just 'barge in' to save the aircraft and those on board.
The expectation is that ATC will do everything in their power to ensure the emergency aircraft is treated with the priority it needs. This is exactly what does happen in most parts of the world. Runways will be closed, other aircraft diverted, or held, aircraft already on approach moved out of the way.
In that case, what sort of spatial awareness (say in an international A380 or a domestic A320) do you have of aircraft in your vicinity? Do you have radar on board that will tell you that other craft are in the airspace you are about to go descend though? I imagine that an on-board 'must land' emergency coupled with an ATC incapacity (for whatever reason) isn't totally out of the realms of possibilities.
We have TCAS, and although that wasn't designed to give awareness of surrounding traffic, in most of the displays it does so quite well. A simpler emergency (that Air France showed us last week), is an engine failure in the cruise. In that instance the aircraft has a problem, though in general it won't be a mayday level call. And you could easily have another aircraft almost exactly below you, or just about to pass below going the other way.....but, once an engine failure happens, you really only have seconds available before you'll have to start descending. It's not a case of choosing to descend...you simply won't have enough power to stay there. So, you'll start a turn with the descent, but that will have to suffice. ATC will probably try to move any other aircraft as well, but the window is very small.
In an emergency descent, you just go...and the big sky theory may have to be employed.
Fuel emergencies can have a nasty habit of simultaneously involving more than one aircraft. In that case, whilst your priority is your own aircraft, ATC do have other issues to work through. But, there is sometimes a disconnect between the urgency that exists in the coughpit, and what is happening on the ground. Whilst pilots historically have no issues declaring emergencies when the aircraft has done something to them, there is a reluctance to do so when fuel is involved...because that contains an element of 'own goal'.
There's the (probably apocryphal) story of the USAF fighter that loses hydraulics and declares an emergency. He's vectored to a USAF bomber base, where he's told that he's number 2 to an emergency aircraft. Thinking they might have him misidentified, he queries this, and is told that they have a B52 making an emergency landing with an engine out....to which the response is "ah, the dreaded 7 engine landing".