The rain affects combat immediately and for each turn that it rains, but the movement effects are delayed.
The CHEMKID map mod was intended as an antique map (matching the original game terrain features) and not specifically a weather map, but in the game system it does change shade when the roads have become muddy. This change is subtle and I missed the change for several game turns in the scenario that I am now playing. My excuse is that, in the last game, the weather stayed mainly clear and the map didn't change, which shows the variability in replays of scenarios.
Here are the two maps for weather changes :
So the moral is 'look at the map', all the information is there, but it is something that I have to keep re-learning, as I just don't take the time to really look.
There is so much in this game which it is easy to miss if you don't take the time.