Optimising autopilot control laws appropriate to the aircraft type, stage of flight, and within safety requirements is a complex design and development task.
The control system is usually programmed to provide a ‘g’ or pitch rate response (often modified by speed and altitude) to satisfy a particular error. Usually there is an error dead-band to prevent the system chasing small errors and creating instability. At the trigger value the response has to be sufficiently quick to satisfy the task and/or pilot perception, yet not too fast to be uncomfortable, and remain within, ‘g’ or rate limits for safety. In addition the aircraft has to remain at or near the correct trimmed condition. Advanced systems might cross couple pitch modes with power or consider ‘energy management’ routines, inertial flight path, etc.
The system will have ‘muscle’ limits so as not to overstress the controls/aircraft and also to provide a safety margin. These can change with task or phase of flight.
In reality these limits are far short of what a pilot could apply, thus when ‘muscle’ limited the system will have to re trim the aircraft.
The trim in particular should be rate limited as it has a powerful effect. Most modern systems are required to be fail-safe, e.g. sensed muscle demand too high, insufficient trim, or trim in wrong direction will disconnect all of the auto system. Similarly the trim will have error dead-band and perhaps a duty cycle – a time period between activity or reversal, to prevent oscillatory instability.
Thus ‘how long a time period to correct an error’ … as above it depends on many variables.
Some of the important aspects about auto flight systems are that crews should not expect pilot like response or necessarily pilot like operation, most do not have the necessary intelligence – perception, knowledge, and experience.
Respect auto flight systems. Don’t expect a super human or quick response in demanding situations. Don’t engage the system with parameter errors (unless the mode is target driven) or particularly with an out-of-trim force, autos can and will bite. Treat them like any other crew member, except that they may not communicate with you as clearly, and unlike a human they cannot indicate that they are struggling before they quit.
Autopilots are like dutiful first officers, but they never learn.