What we're unfortunately seeing in some areas is that the system becomes stressed to a breaking point if too many people become sick at the same time. Either you cannot get through, you are told the ambulance is going to take a while, hospitals are not taking new patients at all, or some hospitals prioritize treatment of certain groups of patients (such as the relatively young and healthy) over other groups (such as the elderly and / or people with existing illnesses).
In the event that official means of getting medical help are unavailable, you might want to try to get hold of that doctor you know, the nurse down the street, anyone with medical training and / or experience. If that doesn't work, depending on the urgency, you might want to mobilize your and the patient's wider circle know you have a patient who is not doing well and that you cannot get help. Ask around for doctors or nurses. Use Facebook? If you have any spare time after that, organize your diary pages, making sure any doctor who has time for the patient can immediately see temperature records, etc. Try not to seem too worried around the patient, because at this point there likely isn't anything he or she is going to be able to do.
If you do get through (by phone?) try to stay calm and help the doctor / hospital assess the situation quickly.
In cases where ambulances are the bottleneck and you feel you need one, you will have to make your own judgement whether you want to try and drive to the emergency room or wait and hope to get through. Plan which hospital you go to first and maybe have someone who is not driving with access to the internet, either in the car or on the speakerphone. Please drive safely in any case, the last thing you want is a sick patient in a car accident.
For now let's all hope our medical professionals can cope with the case load that is coming to them. Do your part in slowing the disease down as much as possible. Let's try to all still be there at the other end of this.