Are there any current statistics on the average time it takes to pass Ugandan immigration at Entebbe airport when requiring a visa on arrival?
I'm not aware of any official statistics but as far as I remember it took me about 30 minutes. The airport is small and didn't look very busy but I arrived at 2:00 am so I'm not quite sure if it's going to be more crowded during daytime. You need to compile an entry paper and pay a fee (about $50 now for a single entry).
It depends a bit on which flight you will arrive from. If you travel Economy on a big airplane (KLM or British Airways) it can take longer than if you come with the smaller Kenya Airways flight from Nairobi. Though for the prominent carriers, immigration typically makes more staff available to handle arrivals.
From my own experience (I've flown in to Entebbe perhaps two dozen times in the past few years), it's unlikely to take you more than 15 minutes, on average.