TL;DR; With theoretical units
Background
It would go probably as the international system went, at the beginning all measurement units were "physical", which meant that when you said "that's 2 meters" the people were comparing a distance against THE meter which was a metal road in France, and so on all of the units, but that didn't worked very well over time, specially for science, in part because of all the copies of the meter were needed with more and more crazy amounts of exactitude, to the nanometers level of scale (about .0000000393701 inches), and because THE meter itself started to change size, just about a few microns (each micron its about .00003937 inches) in less than 50 years it was enough to make a lot scientists go all over their heads and launch a full on global conference where most of the measurement units where stripped down of their physical counterparts and were set to a more "theoretical" idea in which the physical phenomena the unit itself its based in a constant of the universe, for example the meter definition its defined as the length of the path traveled by light in a vacuum in 1/299 792 458 seconds, and the second itself its defined as 9 192 631 770 cycles of a Caesium, as they found out that the earth day/night cycle on which it was originally based changes too.
How it would work
Set into the distant future the only anachronistic thing about the second its that its based in base 60 when everything else its based on base 10, but just to illustrate how much humans love habit, the base 60 comes from the fact that it was what the Sumerian calendar used to be based on, one of the first civilizations to have ever existed, so if it really becomes too cumbersome to use (and it really has to be, if not ask all the people in the united states), it would just be converted to base 10 units and standardized all over the universe, it would go like this:
1 year <- 10 months <- 100 days <- 1000 hours <- 10000 minutes <- 100000 seconds
Sure in earth that would be kind of bothering as a year would be shorter than what today its, but talking in a broader scale it would be way more easy to use, the main reason that the idea of being a whole orbit cycle its so attached to our societies its that it helps with repetition of phenomena, like the stations, but its of no use if you have more than one world.
Ps. Due to relativistic phenomena some sort of FTL communication system would be needed, and due to being humans the conquerors you could set the second definition as 9 192 631 770 cycles of a Caesium within the Earth, Moon, Sol relativistic reference and relay it through the universe, in some places the second will go faster, in others will go slower, but its "universal", actually the gps satellites do exactly that, as their seconds are "faster" they add a few microseconds to their count, other solution its that there could be two units of time, the referential unit, which would be the Earth one, and the relativistic unit that would be the local one
