That’s not how it works though. Before the pilots ever leave their starting airport they load up enough extra fuel to fly to their alternate airport and land there safely. Then they add on the “final reserve” fuel, which is enough fuel to fly for at least another half hour or 45 minutes (depending on airline and region). Together that fuel is called the “minimum fuel”. When an airplane gets down to minimum fuel it must declare that it has reached minimum fuel and is diverting to its alternate. They can only declare an emergency if they are down to the final reserve, and by that time they are expected to have already landed at their alternate. Unless their was a mechanical problem or a fuel leak, failing to divert before reaching minimum fuel would be an error by the pilots, not the ATC. Declaring an emergency just to skip ahead in line, when they still have not even reached minimum fuel yet, would not help the pilot’s career.