It is 100% dependent on the dealer.
Crown Dodge would typically work in a drop-off that same day, and for sure the next day.
Rick Hendrick Chrysler would force me to make an appointment 2 weeks out, even with an ABS light on the dash glowing.
Rick Hendrick Toyota would request an appointment, but would work the vehicle in if you made them.
Bryan Honda/Cadillac has always been a pistol to work with.
Sanford Honda up the road is excellent.
As you can see, I've had two widely different experiences with two dealers in the same manufacturer family (Dodge/Chrysler), and two widely different experiences with two dealers in the same dealer network (Rick Hendrick).
It's really all about who's running the show in the physical building.