It probably depends on the city because in some, the Crash or Paramedic fire truck runs first to a report of an injury complete with all of the medical supplies and fully trained paramedics. The ambulance follows for transport and their medics.
In other cities the ambulance goes alone to non-automobile accident related injuries but is accompanied by the fire department to any car wrecks.
My city has a volunteer ambulance department, though we almost lost it once because public donations were low and good, skilled medics did not want to volunteer plus there were a couple of pay-transport ambulance companies bribing the city council members to get into the business. Since we were one of the very first, years ago, to have a volunteer department, we all kept it, and the city underwrites it and it has grown. They bill you for services under the title of donations but don’t get all bent out of shape if you can’t pay. Our fire department doesn’t run with the ambulance unless called. The police do, however.
That started when it became a ‘fun thing’ to call out an ambulance and then rob it for drugs or shoot the rescuing crews because they were white and in the wrong neighborhood like happened in some cities. Here, at night, the police have to accompany any wrecker calls also, even if your car just has a dead battery.
I had a roomer who liked to drink himself into oblivion every chance he got, which was daily, and twice, fearing that he was in the throes of alcohol poisoning, had him shipped to the hospital via ambulance and the cops showed up first. (The guy even drank my mouthwash, any alcohol based cooking flavorings I had, plus a bottle of rubbing alcohol. I think he was getting ready to start working on the paint thinner when I booted him out.)