FINISHED!!
I built the circuit and after some mechanical and electrical tweaking it works exactly how I envisioned it.
Video of the spinning hourglass:
https://thumbs.gfycat.com/ZestyBlushingCoral-mobile.mp4I've added a few additional features:
- test mode switch: to switch between the 1 minute delay pin and the 15 minute delay pin;
- power switch
- female micro USB: to trickle charge the 3 NiMh batteries (via a resistor and diode)
- solar panel: used as a base and also charges the batteries via a Ge diode
For some reason, I built it with only 3 NiMh batteries in series, which don't give enough power if they are a bit discharged. Will add the 4th to give it more oomph and longer running time.
I am very happy with the result and it works very well and looks quite nice too.
Depending on how it works long term, I might replace the NiMh batteries with a used cell phone battery since it has a much larger capacity and it's easily charged with a small and cheap TP4056 based battery charger board
Thanks again to Ian.M and others for the help!