There's a bunch of ways the leap second can be reflected. :59 twice, :60, leap second smearing (adding small amounts of time leading up to midnight). How your particular system reacts depends on the use, how it's configured, etc.
That makes sense. I was being slightly facetious, and I think most devices would NTP re-sync after some interval. But it would be cool if my phone did in fact show 3:59:60 (I'm pacific time) and/or analog smartwatch paused on the 59 for two seconds.
Have our timekeeping protocols really accounted for future events like that? It's my understanding that NTP really only synchronizes time periodically.
Your phone/computer should (according to the standard) know when the leap second is and apply it. NTP etc. transmit the information about leap seconds well in advance.
It is possible your device may not see a leap second, only eventually observe that it is running one second slow, and smear to correct for that.
NTP includes flags for "leap second at the end of day", so devices that check NTP time at least once a day get the information. The systems internal list of leap seconds is possibly updated before that through normal system updates.
Only if you are in UTC timezone I suppose
It is unclear for timezones behind UTC. If you are in UTC+1, the leap second happens at 1 am, so the countdown would be over: no fun here. If on the other hand you are in UTC-1, the leap second happens at 11 pm, so if the countdown clocks are not corrected for the leap second, friends can still be annoyed.
Either way, your friends will be annoyed..
What clock is set that precisely, hasn't drifted since being set, and won't pick up the correction by itself?
A GPS watch without a firmware update to handle the latest leap second?