What time zone will the crew keep?
This is an important question, albeit not yet frequently asked. It presents a collection of conundrums for the Artemis Project, and for anyone who wants to fly humans to the moon.
During STS-88, when we coupled the International Space Station's Node 1 ("Unity") to the the FGB ("Zarya"), mission controllers in Russia had difficulty converting from Mission Elapsed Time to their local time. The complained about this bitterly throughout the flight and there was even one incident where they were two hours off on the time they promised to provide some vital data to the flight controllers in Houston.
These problems are exacerbated in the Artemis Project by our plan to have mission support centers in a globe-spanning string of facilities, with viewing rooms open to the public.
Once a Space Shuttle clears the tower, NASA reports mission timelines in Central U.S. time; in other words: Houston time. (Zone -5.)
This makes life easier for the astronauts (who live in Houston) but it's mainly for the several hundred people who support the missions on the ground. They have enough to think about without having to calculate time zone changes in their head.
Inside the Mission Control Center, all our engineering data are tagged to UCT, even the test runs in the laboratories and the water tanks. The time tags are corrected to Ground Elapsed Time. GET is how much time has passed, on the ground, since liftoff. During a long space mission we can accumulate enough errors due to relativity that it would really mess up the computers. The computers are happiest if they can stay in synch within the boudaries of a single simulation frame -- 25 milliseconds -- and become rather petulant when they can't agree on what time it is.
Moon flight complicates the clocks
Flight to the moon on a typical Apollo trajectory has an unfortunate interval between launch and lunar orbit insertion. You either have the crew awake in the middle of the night, according to their circadian rhythms, or you have them trying to get to sleep after a 19-hour day.
Neither of these is a very good solution for a mission where single blunder can result in disaster.
The reference mission timeline gives a breakdown of the critical events between launch and landing. If you launch at 8:00 AM and follow the Apollo 16 mission profile, you do Lunar Orbit Insertion at 3:30 in the morning.
We don't know
Because of all these factors, we have not decided what time zone the crew will use, or when the major mission events will be scheduled. The most likely choice will be to schedule the landing on the moon to maximize the audience on Earth. This ain't Apollo; we're in show business.
Our reference mission is designed to loiter in Earth orbit for as long as needed to assemble and check out the spacecraft. That gives us a lot of flexibility in scheduling the departure from Earth orbit, and hence arrival at the moon. The spacecraft also has the option of loitering in lunar orbit until the crew (and the audience) are ready for landing.
So, we really don't know the final answer to this question. It depends on how the Artemis Project develops, the crew's schedule, and where the audience is.