As an ex D&D controller I've had several of these for real (as well as dozens of practice ones!).
If your compass is U/S what difference does it make what heading you fly, you're only interested in pointing in the right direction, as are ATC.
ATC will use what they see on the radar and (in my experience) will use the No Compass No Gyro procedure. Although it does seem that civil pilots are not familiar with this, so.....
"Adopt the No Compass No Gyro procedure, make all turns rate 1 (thats 3 deg per second), start and stop turns on the executive word NOW!"
With a standard rate 1 turn you can time turns, i.e. 30 deg = 10 seconds (stopwatches supplied! ) but its generally been my experience that its best to just watch the aircraft turn, taking into account the lag on radar (about 30 degrees).