While possible, I dont think the effort will pay off by trying to use a microcontroller and neopixels here..
I made fully functioning Arduino compatible circuit, designed to fit in the face of a watch for the TASM webshooters: (thread is here somewhere)
ie:
These were minimized to ensure they would fit in the prop.. (also had SD card sockets on the bottom for audio playback)..
You would have other obstacles here.. such as voltage regulation. (Neopixels require +5v.. but more current than an Arduino can provide.. so needs to be powered directly from a stable +5v power source)..
You would still need to create a custom PCB.. (unless the watch is big enough to fit a Pro-Mini or something inside of it?)
Questions:
1.) Do they need to change color? or will they only be one color per watch?
2.) Perhaps 'faking it' will work? Such as get some thin acrylic cut into circles.. with an etched 'ring' around it. Then you can throw an led or two along the edge to 'edge light' the etched ring?
Neopixels come in smaller packages than just the 5050 sizes.
I believe there are also 1206 RGB leds too? (I'd have to look what I have to be sure... or just ebay/google it)