I always found this type of watch to be facinating. The design goes back to the Magician Robert Houdine. Just re watched
Hugo and there is a call out to the 'mystery' clock.
The main issues with battery was that at the time small batteries used mercury. Which gave a lower voltage than silver. Lithum is a more recent production. The natural voltage is different. Mercury is 1.35 where silver is a bit higher around 1.5. Lithium is closer to 3V.
I did do a quick search to see what caliber this takes. I see this is the elusive
Scarface watch.
I found a photo of the inside of the watch
So there is not really a movement per sea as it is pretty much integrated into the design.
There is a thread that the search turned up on these forums
https://omegaforums.net/threads/1-261-produced-omega-la-magique-de-ville-18k.46379/ which give some details including the caliber number 1357
. I do not see that number in the usual datasheet download sites. I did find the 1355 which is a quartz tuning fork movement.
Mostly I play around with the 1342 quartz movements. I have some background with micro electronics (I design pipe organ relays.) The main problem is that the chip factories used to make the time control chips have long since crumbled to dust and the designs shredded.
In theory a modern general purpose micro controller could drive the stepping motor. These can be quite small. Typically though they run at a higher 1.87 voltage up to 3v or so.
The stepping coils can also have issues. The wire subject to mechanical stresses. Or even over currents from back emf and static discharges, which were not as well known back in the 1970s.
It also looks like these often were in gold cases which would trend a few of the examples to the fate of the fiery furnace. Although that could lead to some of the movement parts floating about.
A lot of repair of the early quartz does involve cannibalizing what is floating out in the wild.
I think though with the
Scarface connection that there are specialist who do focus on hoarding these parts and servicing what there is that can be serviced.
I doubt there is much Omega can do since the institutional knowledge of this sort of thing is now lost. They probably do not have the parts. So hopefully you can find an independent who specializes in these.