Just dl'd to take a look.
This is an scaleform UI mod. However it gets the current time from the computer is not accesible to papyrus as yet afaik,
You sure? Ah well, worth a try.
I wasn't planning to make the clocks sync up perfectly, but if, whenever the time is shown, it could update a record somewhere which is readable in Papyrus with the time as of last loading screen, that would work (to within a minute or so)
Basic idea:
1. Kenmod shows time, copies time to some file/record. (whatever is possible)
2. After the loading screen (which presumably can be detected, as the game autosaves [if all else fails, update time on autosave]) the game clock is set to the time given by the record.
3. Repeat.
As the timescale is 1:1, this would give a pretty accurate time. Of course, if you wait in the pause menu for ages, without going through a loading screen, you'd be out by that long (but I'm hoping that people don't mind that, compared with "it is night at night, day in day, morning in morning, etc.". Vanilla game doesn't have any clocks, right?)
...Now explain to me precisely why I am wrong and this will not work

this is my favourite kind of speculation: entirely uneducated.
If it might work, I might make this in a couple of weeks. I was meaning to learn the language, anyhow. If it won't work... I get to learn even more.