Normally there is a correction component applied from a fluxgate sensor in the tail ,but in your case this is just a gyrocompass, so the correction is simply dialled in, and the compass is used, along with regular reference to the magnetic compass, to enable the aircraft to make precision turns, and to navigate with no visible conditions. You turn on the gyrocompass to a heading, then align to the magnetic compass, to make a final correction in straight level flight, because the compass will have a turn error. Compass will swing and have positional error in a turn because of the banking you do to turn, but the gyrocompass will simply ignore most of this ( there are edge cases and turn limits) enabling a precision turn. Then you simply keep correcting the 12 degrees per hour drift ( there will be an extra component because of the rotation of the earth around the sun, but this is not sensitive or low noise enough to pick that up, nor the rotation of the sun around the Galaxy, or the relative motion of the Local Group) every hour, using the magnetic compass, and it runs from there fine, as your course will be not a straight line but a sector of a Great circle anyway.
Incidentally rotation rate is independent of position on the planet, all places rotate 360 degrees every 24 hours roughly, just that some places like the poles have a lower angular acceleration. The gyro erection system, which keeps the gimbals in a horizontal plane in level flight ( and which take a while to erect it unless you use fast erect by pulling or pushing the adjustment dial) will also apply a torque that tends to keep it pointed in the same direction.
OT I really wanted to have a moving map display like I used to work on, but that was a 50kg box, needed it's own 30kg computer to drive it, and used a pair of gyro blocks ( redundancy), a fluxgate sensor, an air data computer to give physical inputs of speed, altitude and such, and which needed around 5kW of power and cooling to operate. Those were mechanical marvels, like the older analogue autopilots and electromechanical navigation and aiming systems were.