How much lag does an Attitude Indicator experience?
There is no virtually no lag on attitude indicators. The traditional ones are based on a mechanical gyroscope. Due to the gyroscope being gimbaled, it remains in a fixed attitude. 

When the aircraft changes attitude, the gyroscope in the attitude indicator remains at its original orientation. When you look at the attitude indicator you are looking basically directly at the gyroscope, packed in a nice horizon displaying wrapping. 



The modern attitude indicator in a glass cockpit gets the attitude information from the Attitude and heading reference system (AHRS). There is of course some lag associated with digital data transfer and display refresh rate, but these effects are too tiny to be noticeable by humans. 

The first generation of these systems used mechanical gyroscopes as well. Nowadays most AHRSs uses Ring Laser Gyros or Fibre Optic Gyros. These gyroscopes use relativistic effects of light travelling clockwise and counter-clockwise through a circular path. The accuracy of these systems is very high and there is no delay.

Another technology that is used is are Micro-electromechanical Systems (MEMS) gyroscopes. The are based on a vibrating structure. The direction of a vibration tends to remain in the original direction, even though the supporting structure rotates. MEMS are relatively cheap compared to RLG and FOG but they are less accurate. Due to developments in the automotive and mobile telephony industry where attitude sensors are used for various applications the accuracy of MEMS gyroscopes has improved while their costs have decreased. This makes them now interesting for application in avionics.