I've had to look in several references to piece an answer together, but I
THINK I've got it right. I'm willing to be corrected, though.
Firstly,
this website definitely says it measures about the vertical, and I don't see how the turn rate would check against compass and stopwatch if it didn't.
Pallett (
'Aircraft Instruments' by E J Pallett) agrees with Bookworm. To quote, he says
'In practice the gimbal ring deflection is generally not more than 6 degrees, the reason for this being to reduce the error due to the rate-of-turn component not being at right angles to the spin axis during gimbal ring deflection'. So he is saying that the gyroscopic effect is measured about the yaw axis, or at least, not more than 6 degrees from it, which is acceptable.
But, hang on, depending on your TAS, a Rate One turn could be 30 degrees of bank or more. This is where the pitch precession comes in.
If you banked to, say, 30 degrees and yawed, the axis of yaw is no longer perpendicular to the horizon. So you would be yawing towards the ground, if you took no corrective action. This is why the pilot applies back pressure during the turn and, if necessary, increases power as well. In doing so, he is altering the pitch attitude of the aircraft relative to its yawing axis.
This pitch produces a second precession, as LOMCEVAK says. In a rather elderly version of RAF publication AP1234 which I have in front of me, it calles this 'Looping Error'. It then goes on to say
'The indicators are calibrated for rate 1, 2 and 3 turns at specified angles of bank and TAS, and looping error is taken into account in this calibration'.
So I think that the answer is that the gyro actually measures with respect to the yaw axis (or within 6 degrees of it) but the combination of looping error and linkage to the needle gives a read-out of rate of turn with respect to the vertical.
But what a hell of a question! I'm still not sure what answer the JAA want.