After a bad earthquake in California many years ago, citizens started preparing themselves to be ready in case of another disaster. During that earthquake, emergency services were overwhelmed and could not reach all emergencies. This gave rise to the Community Emergency Response Team (CERT) training.
I went through some of that training. It would make sense, in your situation, for citizens to train and prepare to handle basic emergencies until the professionals arrive. In many cases, it is simply about having a plan on how to handle different kinds of emergencies.
Another good thing to have is radio communications. Mobile phones work well. But you can only talk to one person at a time. With radios, many people can coordinate together to handle emergencies. California also has radio networks as forest fires will often destroy cell towers. At the very least, with radios they can talk to each other locally to coordinate when phones don't work.
I think that if you can't rely on your government, you may not have any choice but to organize your community to manage for themselves.