Atomic clocks do not deal with leap seconds and the length of a day. All atomic clocks internally just keep a counter that indicates the number of seconds (or rather picoseonds) that have past since some universally agreed point in time in the past. Mid-night on January 1st, 1970 in Greenwich is such an universally used common time point in the past. Time synchronisation between atomic clocks is also based on such a standard.
EDIT: the below description for computers is correct for leap days, time zones, and such. But apparently I assumed too hastily that it applied to leap seconds as well. Leap seconds are actually incorporated into the running count on some computers. Thanks for Ilmari for pointing that out.
All clocks in computers also just keep a running counter of number of seconds since a fixed time point. They then synchronise that counter over the Network Time Protocol with the internet to keep the clock in sync with other clocks.
It is only when translating those counters to human readable times, that leap seconds (and leap days, time zones, and summer/winter time) come into play. We humans like to have the hours, minutes, and seconds that we use to be correlated to the spinning of the Earth, such that the sun is always directly above at 12:00. For that reasons, computers translate the internal clock counter to a human readable time, according to mapping tables in the operating system. Those tables also take care of all the different time zones in the present and both the past and future.
Those translation tables need to adapted when the Earth spins slower. But that is not a dramatic change. If it is agreed in advance when a second is going to be skipped instead of added (a negative leap second), then those tables can just be adjusted in advance without major consequences.
Sattelite navigation systems
For satellite navigation systems, they are not affected by this mapping at all. The clocks in navigation satellite don’t deal with human times either. Only two things are important for the satellites:
- that each satellite knows precisely above which geographic location it is located at any point in time.
- that the clocks in the satellites are synchronised with each other, such that it is guaranteed that they all show the same time from an earth surface point of view.
Due to general relativity, clocks on satellites run faster than on Earth, and need to be compensated for that. Satellite orbits can also not be predicted with perfect accuracy. For both these effects, there are already adjustment mechanisms in place. Both clock speed and the orbit information of the satellites are regularly synced with ground stations in order to make sure the information remains accurate. A milli-second more or less is not going to upset those adjustment mechanisms.