Talking about vacuum-driven gyros, which are what I'm mainly familiar with: When everything is working as designed (sufficient air pressure for normal gyro rotor speed, moderate maneuvers) there is some lag.
You'll notice it if you're looking for it, but the "1-2 seconds" mentioned in the linked answer seems on the high side (at least based on what I've seen in light aircraft).
Vacuum-driven attitude indicators lag because of the way they operate: Basically a set of vanes are always trying to pull the gyro upright relative to gravity, and the inertia in those vanes and the air flow induces a slight lag before the rotor (and visual indicator) moves.
That's not the only error in your trusty attitude indicator. Stealing liberally from an old edition of the IFR Handbook for the rest of gory details details:
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Another group of errors, associated with the design and operating principles of the attitude indicator, are induced during normal operation of the instrument. A skidding turn moves the pendulous vanes from their vertical position, precessing the gyro toward the inside of the turn. After return of the aircraft to straight-and-level, coordinated flight, the miniature aircraft shows a turn in the direction opposite the skid. During a normal turn, movement of the vanes by centrifugal force causes precession of the gyro toward the inside of the turn.
Errors in both pitch and bank indications occur during normal coordinated turns. These errors are caused by the movement of the pendulous vanes by centrifugal force, resulting in the precession of the gyro toward the inside of the turn. The error is greatest in a 180° steep turn. If, for example, a 180° steep turn is made to the right and the aircraft is rolled out to straight-and-level flight by visual references, the miniature aircraft will show a slight climb and turn to the left. This precession error, normally 3° to 5°, is quickly corrected by the erecting mechanism. At the end of a 360° turn, the precession induced during the first 180° is cancelled out by precession in the opposite direction during the second 180° of turn. The slight precession errors induced during the roll-out are corrected immediately by pendulous vane action.
Acceleration and deceleration also induce precession errors, depending upon the amount and extent of the force applied. During acceleration the horizon bar moves down, indicating a climb. Control applied to correct this indication will result in a pitch attitude lower than the instrument shows. The opposite error results from deceleration. Other errors, such as "transport precession" and "apparent precession," relate to rotation of the earth and are of importance to pilots and navigators concerned with high speed and long-range flight.
A more common cause of a (noticeable) gyro lag is low vacuum, which results in low rotor speed and makes the entire instrument sluggish.
High vacuum can cause the opposite effect - excessive rotor speed, and "overshoot" in maneuvers.
You can find more of the old IFR handbook over on pilotfriend.com. "IFR Exam-o-Gram #24" available here is also a good read - both of these are from the days when the FAA expected pilots to cram pretty extensive systems knowledge into their brains.
The gyro vanes picture above is out of the Airframe mechanic's handbook - AC 65-15A which is actually still current.
