You just need some fake number to maintain the time.
If your clock works by minutes then there are 1440 minutes in a day. Make it an integer/float and add some number to it per second which translates to game minutes per real second. Reset it to zero if it ever equals 1440 or more and start the count all over.
Now you know what time it is in the game, so you can correlate that number to some value on your weather system. By dividing, multiplying or normalizing that value you can turn it into other numbers that you could use to adjust rotations of a directional light or light intensities/colors for instance.