A traditional Gyro does indeed tend to maintain it’s orientation relative to it’s beginning orientation, so as we fly from one location to another it’s vertical axis remains unchanged. So a 1200 Nautical mile journey would produce a tilt of 20 degrees ( 1 degree per 60 nautical miles). As a result early simple gyroscopic attitude and heading indicators needed regular corrections input thru an adjustment knob on the face of the instrument. In addition these instruments suffered from errors induced by bearing drag and accelerations. They were adequate enough on small aircraft during short flights.
Modern jet aircraft use much more complex systems for attitude (pitch and roll) intruments that are not directly driven by gyroscopic mechanical output. These can be generally divided into two types, Inertial and AHRS (Attitude Heading Reference systems).
INERTIAL - In the 1960s Long Range Aircraft flying across oceans particularly began using Inertial Navigation Systems which were originally developed as extremely accurate missile guidance systems. Using very accurate gyroscopes, accelerometers and math equations that fine tuned output, these systems could dead reckon navigate without external sensors and as a side benefit output pitch, roll and true heading information to flight instruments.
Another side benefit when receiving information from air data systems they could compute wind direction and speed. Early model 747s usually used three of these systems for accurate oceanic navigation. I flew many oceanic crossings with the navigation error as little as one mile after a oceanic crossing. Very amazing stuff at the time! Though still in use, INS has been superceded by Inertial Reference Systems which work basically the same but don’t have a navigation function specifically built in, that job is done by the FMS using position and heading outputs from the IRS.
Mchanical gyroscopes have been replaced by much more reliable Ring Laser Gyros. With no moving parts these amazing devices use lasers directed around a course with mirrors, two moving in opposite directions where the difference in beam timing along with accelerometers produces information to compute pitch, roll, heading and movement!
Whether INS or IRS, both systems require a very accurate starting position to be input when first turned on. This position is then used to determine the earth’s rotating velocity at the initial position based on Latitude. The system then goes thru an alignment stage lasting from 6 to 15 minutes as it tunes its internal sensors. Even small errors in starting position will effect its accuracy over time.