Q.  How are the barrels rotated?

A.  The rotational force is delivered by two hydraulic drive motors operating off of both main aircraft hydraulic systems (A+B).  The motors deliver their torque to gearbox in the Hydraulic drive assembly.  This in turn delivers the force to two drive shafts.  The forward one goes to the rotor assembly on the GAU-8 itself, while the second delivers it's power to the ammunition drum.

Q.  How long is typical spin-up time?

A.  While the system is capable of operating at 4200 rounds per minute, it is limited to 3900 while installed in the aircraft.  Since the GAU-8 has 7 barrels it needs to spin at 1/7th of the firing rate.  Basically since it's rotating a touch over 550 rpm the spin-up time is almost instantaneous. 

On another point the statement on the gun gas on affecting the engines is complete rubbish.  During the firing of the GAU-8 the engines ingest so much gun gas that the engine igniters (think super spark plugs) must fire to ensure a stall does not occur.  The engines also will ingest so much soot from the firings that they need to be "water washed" (have water and a mild cleaning solution misted into the engines while running at approx. 80% power) periodically to remove soot from the blades.  Anyone who has had to clean an A-10 will tell you the gun gas goes just about everywhere aft of the nose.