I'm reminded of this delightful site featuring an HP brand atomic watch with nylon band: http://leapsecond.com/pages/atomic-bill/

Somebody already made one - Hoptroff, a London-based watchmaker produced a pair of wristwatches based on the CSAC a few years ago. Don't bother searching for a price or where to buy; they've since pivoted from making watches to being a time-as-a-service software company.

IMO it was a bit of gimmick. Apart from being a little too chunky for a wristwatch, just because it's an "atomic clock" doesn't mean it's "atomic powered". IIRC the CSAC uses about 1/8 W which is really pushing it for a low-power device like a watch. It might lose "one second per millennium" as advertised but good luck keeping it continuously operating anywhere near a thousand years.