Multichronometers, have both local time synced to the local system that is generally used when in system, and Imperial Standard Time. It's appeared in multiple supplements to date, but no Mongoose supplements yet as far as I'm aware.
It's not hard. An atomic clock or the 52nd century version of it is going to be standard equipment for Imperial locations, and while there are very, very, very small discrepancies, if you are just needing HH:MM:SS time your ship can synch its timeclock at any starport or orbital landing sat controller.
Your personal comps are going to be based on whatever time you want. Your ship may follow it's own cycle, and you'd have your planetary local time, or IST. Though unless you are doing something that requires IST, local time / ship time is going to be your guide for all your activities. Ship crew MIGHT run on a 24hr clock, they might not - depending on the ship and crew and what is the norm. Stations may have their own time periods and be different than the planet they orbit - if it makes enough sense. A planet with a 14hr day might now work for a station, nor would a planet with a 37hr day,
Unless it's key to an adventure, stay with a 24hr clock since it's far easier to keep track of in game.
I never thought to lookup Office of Calendar compliance and I wouldn't have guessed there where fleets of ships assigned to this office.
Some thoughts on how time is sync'd across imperium space. One of my ideas where devices connected by quantum entanglement (or whatever that is in OTU) that will sync over vast distances.
IAT (Imperium Atomic Time) is the standard time reference in the Imperium of 24 hrs.
Each system broadcasts 2 time signals, local time of the system (it could even broadcast the current time for all worlds and moons in the system) and Imperium time. Any ship that jumps in or or out the system would use this to verify the expected day and time they calculated while jumping into the system.
The central atomic clock on Capital would be in orbit around the planet and is called a Deep Space Atomic Clock (DSAC).
Some sort of Universal Time Standard would need to be developed that allows the other space fairing races to coordinate there standards of time. This really would only be used for diplomatic missions.
I am curious what the applications of standardizing your clocks across interstellar distances is, given you can't even control how long a trip from one system to the next is with any real precision. Grand Admiral Arbellatra was about to get 60 ships to all arrive in Capital within an hour of each other by dint of a lot of drive sync-ing drills over the course of the months it took to travel from the Marches to Capital. Less elite and dedicated forces are gonna be far more widely dispersed.
Its not like when the Japanese tried to time the war declaration with the US to be slightly before the attack on Pearl Harbor. If the Zhos mention a new Frontier War to the Duchess of Mora, she's still gonna need a week + to get the word out to the nearby systems. So you don't need to be *that* precise with your timings.
From a the Imperium is run by bureaucrats, most likely very important to them.
Fleshing out the OTU, might be nice to know.
From a thought exercise, really interesting to think about how it might be done. We have managed to quantum entangle photons and change there color over a distance. What stops the OTU from using this tech to do say time syncing or morse code style communications over great distances. When the game was developed none of this was even know or thought of.
Syncing clocks assumes minutes or seconds matter to what they are doing. I just don't know of anything that actually has that level of precision relevant across interstellar distances. The fastest two things/events/whatever can interact in two different solar systems is the speed of jump 1.
It would be different if there was some kind of ansible tech, but there explicitly isn't. That is practically the defining characteristic of Traveller settings. Even when they use stutterwarp or zimm drives or whatever instead of jump drives, they don't have ansibles.