If there's trees on houses or structures they will come first. You wouldn't want a tree service performing maintenance trimming on your neighbors trees while there was a big oak laying in your living room. So we usually put the most important first. Then we move to fences and empty sheds. Then trees that are down but not on anything. Once all of that is done then we get back to our list. That may take a week or 3 months depending on the amount of damage and number of jobs. I do however make calls and explain the situation to the customers even if I can't give an exact time to start the job. Sometimes I'll put off customers to do storm damage and while I'm at that job the neighbor comes out and wants his trees worked on while I'm there. Sometimes I could work an entire neighborhood that way and be there for a month. I explain that this is an emergency job, leave them an estimate and try to get back to my regular customers. They rarely ever call because they want their work done right then. But I keep my regular customers who have been patiently waiting. That also weeds out the ones that want it done yesterday.
If you can't get in contact with them to see what's going on, I'd find someone else. Just let them know that in a voice mail, email or text so they don't show up 2 months later to a completed job. No matter how busy I get I always return my calls that night or at least have my wife do it the next day. So there's no excuse for them not letting you know what's going on.