You don't have to worry about scaling- Earth has one moon. And there are just eight planets in the solar system. It seems obvious that any off-earth colony g

You don't have to worry about scaling- Earth has one moon. And there are just eight planets in the solar system. It seems obvious that any off-earth colony governed by humans on Earth will have at least one time system that attempts to reference Earth time. There may also be a local time scale in sync with the rising or the setting of the sun. But that probably won't be the case on the moon, because a "day" is weeks long. So the humans will use a clock that references UTC. Perhaps a fixed offset from UTC if the Moon base is governed by a single country. The hard science part is dealing with gravity wells and keeping stuff in sync. But the top level answer will be that clocks will follow UTC, and any communications systems between the Earth and the Moon will be built to tolerate 2.5 seconds of lag.