I am unsure of how your healthcare system works. Is it possible to pay for more timely service from a private hospital or private doctor?

If it is possible but not feasible because of cost, then an American, speaking practically, experiences the same thing. Not enough money to pay for medical service. However there is an important difference.

An American must pay something out of pocket. The financial burden may lead to bankruptcy and ultimately no medical care at all. I've noticed that you did not raise concerns about bankruptcy but instead your concern was over slow medical service.