I built the electronics and wrote a very simple ATTiny85 program for an automatic watch winder some time ago. (The mechanics and all assembly was done by the guy I made it for.)
It was much simpler, without a display or other fancy stuff, and simply used a geared 12VDC motor to rotate in one direction for 5 minutes, rest for 20 minutes, then rotate in the other direction for 5 minutes, ad infinitum.
The output shaft of the geared motor ran at 3RPM, and a tin can was used for a level of protection from the magnetic field of the motor.
So I'm with MorganS - a geared motor is really the best and simplest approach. My friend got a suitable one from AliExpress for a few dollars.
For simplicity, in the circuit I used a DPDT relay to achieve the reverse effect, and a small (logic level) power MOSFET for the motor power on/off control.
This is the circuit:-
Edit: Oops, almost forgot - here's a link to a video of it in operation:-
Homemade automatic watch winder
And the circuit board, in case you're interested. (The same could be done using Veroboard or similar.)
