How long does it take for a propeller engine to spool-up?
In my Cessna Cardinal, with 360 cubic inch (~5.9L) carburated engine, I push the throttle in and the response is immediate.  Just like stepping on the pedal in a car with a standard transmission because propeller RPM = engine RPM. I have a constant speed prop, with controllable pitch angle, so it sort of acts like a standard transmission, I can dial it to any 'gear' I want to. For take off and landing, the pitch is flattest, and most responsive. Just like in a car, I can dial the prop pitch flatter and add more throttle for a climb from level cruise, then when at altitude I can dial the prop pitch up and throttle back down for efficient cruising. In planes we generally don't go to full thrust instantaneously, that's rough on the engine, but smoothly push the throttle in, maybe 2-3 seconds from idle to full throttle.  During landing, sometimes a bit of throttle is added to arrest an increasing descent rate, and even that is applied smoothly, and rarely full throttle unless a go-around is needed.

This link shows how the oil pressure, springs, governor, and other bits are used to control the propeller pitch when the RPM has been selected.

http://www.boldmethod.com/learn-to-fly/aircraft-systems/how-a-constant-speed-prop-works/