In the U.S. the pilot would simply declare an emergency which gives her the ability to do anything she needs to do to meet that emergency. If ATC has an objection, they will voice it but the pilot in command has the final authority, not ATC. Later, there may be an investigation and the pilot might be found guilty of breaking an FAA regulation. The FAA cannot inpose any criminal penalties and is limited to fines and license suspension or revocation. so the consequences of disregarding ATC are fairly minimal. In any case, it is very rare for the FAA to take action if there is even a strong argument that the pilot took inappropriate emergency action. If I were overflying Groom Lake and needed to land, I would simply declare an emergency and land. I don’t think the FAA would care. There could be other agencies who may prosecute me but I’m not going to worry about that when I need to be on the ground.
In other countries the rules are different but in reality, if I’m overflying Cuba and I have a fuel emergency, I’m going to land at the place I feel will keep the passengers the safest. If that means landing at an airbase or some restricted facility, I’m going to land and work out the details and violations later. Better to spend some time in the brig than have 150 deaths to account for.
The questioner asks about a low fuel situation. In those cases, ATC is exceptionally helpful in assisting an aircraft in need of an immediate landinng.
If ATC, cannot provide separation from other aircraft, they will deny a clearance, even to a low fuel airplane. It’s then up to the pilot to make a decision on the safest way to proceed. It could be to go ahead and land or it could be to take some other action. The runway may be under construction so ATC will not issue a clearance. The runway may have a disabled airplane on it. There are lots of reasons that ATC will deny a clearance. I will note that in my 30 years as an airline pilot, I’ve never observed an “oblivious” controller. They will make mistakes just like I do but they have always been attentive and eager to help.
It should be remembered that ATC doesn’t give permission, it offers clearances. If a pilot is offered a clearance, she may accept the clearance or reject it. If she rejects the clearance, ATC will offer an alternative. If no alternative can be found, the airplane has the option of using emergency authority or taking some other action that may not require a clearance. The role of ATC is to provide separation from other aircraft. While they are amazing professionals who help navigate, deal with emergencies, separate us from mountains, try to keep us out of restricted airspaces and a whole lot more, it is important to realize that ATC is not FAA police and do not have any role other than keeping us from hitting each other. It falls to the pilot to keep from hitting the ground, navigating and staying out of restricted airspace. Just because a pilot has a clearance to do something doesn’t make it legal. If you recieive a clearance that takes you somehwhere you shouldn’t be, it’s your fault, not ATC’s. It is the pilot’s responisibility to comply with the regulations not ATC. I’m really not trying to minimize the role of ATC but I think the word “controller” tells folks that they control the airplanes and that’s not true. They assist the crews by giving them instructions that keep them away from other airplanes. It’s a very important job and people should be proud of Air Traffic Control in the U.S.