Hi Christine
I have been living in Kampala since a bit more than a year now, and prior to that was a frequent visitor. I can share my experiences, that especially he areas of Kololo, Nakasero, Naguru, etc are quire safe and you can move around at any time of the day / night.
If you are moving around in a car, then you have nothing to fear. WHile if you are moving around on a bike / walking, obviously you need to excercise usual caution.
I have moved around on work, as late as between midnight to early morning, and have not encountered any issues.
I can also mention about Mbuya and Bugolobi as quite safe, apart from Kampala road or Entebbe road.
Parts of Entebbe are quite dark but nevertheless I have not encountered any reason to worry.
Hope this elaborates and clarifies.
BR/GG