I worked at NPL for 11 months, where they have one of the 11 6 atomic clocks worldwide. This machine isn't a clock at all, but rather a machine that fires atoms back and forth and measures their travel time. It is synced with the other 10 worldwide and if any of them go out of sync, it flags with their monitoring system. It's virtually impossible for all 11 to fail due to the amount of redundancy on each site. The time is then broadcast via rather primitive radio signals to these "radio controlled clocks" which then automatically adjust.
It's a fascinating system, some of the most advanced technologies in the world maintaining our time, and broadcasting it via a system that's over a century old.