Hi
There are two clues that make me think there is a missing part. Maybe just a nut.
1. there is a round pin on the shaft that I assume engages the keyway on the outer plate.
2. There is a shiny mark just below the teeth on the gear wheel.
Could this indicate a friction clutch surface engaged there, or just the back of the outer plate. Is it relieved
engaging on the outer edge only?
It is in the right location for maximum friction torque to be applied.
Maybe the plates were simply clamped together after setting with say a wing nut or knurled nut.
Once set the only way to change the time would be to access the mechanism directly.
simple clamping is all that is needed.
Image below.
I have marked the areas in red.
Regards
John