I... can't say I have any experience with MUD engines in particular...
... but in theory, it does not seem impossible, or even that difficult?
Basically, just have a preset world space that is cordoned off into strips, bands, and that functions as essentially an overlay layer.
Split it into 24 slices, give each slice an offset from "UTC" ... every time the ... "UTC" of this world rolls over another hour, iterate each local time zone between... sunrise, morning, midday, evening, sunset, night.
Something like that basic framework.
Hell, you could also vary the length of days, of each of those 'lighting condition states' by latitude, with another set of strips, and then use both of them together to come up with some kind of 'season' offset.
To maintain your own sanity, if trying to travel long distances or something, just convert everything to 'UTC' time, then calc travel time, then calc back to local time once you've arrived.
... And be wary of crossing the 'international date line'.
Localized weather is... well, it can be as simple as just having like, another grid splitting up the world into weather regions, and then you have something like a 'climatic zone index' that you assign to each grid square, and each climatic zone has its own lookup table of possible weather states, which have some weight value associated, and then just every hour or 6 hours or day or w/e, you randomly roll for a new weather state.
You could also get much more complicated if you want to simulate things like moving pressure systems that interact with significant terrain elevations, moving humidity patterns, ocean temperature circulation patterns, wind speed, etc...
I... am assuming this is either 2d or just like ascii character based?