Also on my Cadillac 1956 the original clock had stopped working and I decided to disassemble and see if it was repairable.
I soon found the trouble. It was a small spring lever that transfer the turning motion from the driving spring mechanism to the clockwork that was broken. I fixed it (hopefully for a long period) with a thin steel wire that I glued to the axle and lubricated the clockwork connected a batery charger and heureka! it worked. After a few days it still winds itself up and works. I´ll keep you posted of the development and if it was a lasting fix. Anyway I now know the reason for the clock not working when I got the car…..
It didn´t last long before it suffered a coil melt down…had to repair or convert to a modern kind of clock…. see Link