I am not an elec engineer and don't know how digital watches are designed in reality but I suspect that if you want to design it yourself a textbook or resource on "digital logic", "digital logic systems", "digital systems" would come in handy and be an ideal starting point. Check out a good library, at a university especially, for what the popular titles are.
I recall designing the logic for similar sorts of things in class. We weren't often concerned with realisation however.
Hoop up some logic-gate-ish circuitry (I don't know how this stuff would be packaged for use in a small watch... In reality it's probably done via programmable logic device
things which is less diy-friendly. But it is doable in a simpler manner with discrete ICs
) to a few 7-segment displays
, a battery and a quartz crystal oscillator.
Design some sort of counter that triggers the displays appropriately and counts via the crystal. I assume this is how watches work..
I have only a faint recollection of this stuff now. I could re-read my notes... meh.
I'm not qualified to go into detail sorry. The book will explain everything I've mentioned - and left out.
I believe the process for a basic project involved some state-transition diagram
... do something else... get a circuit diagram... hook up some logic gates and some circuit... voila.
You won't need to know much about circuit theory to do something primitive and probably adequate for a first go, just know what a circuit is.
Reading an appropriate basic digital EE book requires no pre-requisite skills you wouldn't have already. If you have an inquiring mind it will be easy.
You can possibly buy counter ICs (integrated circuits) designed specifically for running 7-segment displays. I have no idea...
If you want to learn from the ground up read one of those books.
I'm sorry for a less than stellar response but I hope it can be of help and get the ball rolling.