I am not a descriptive expert but I think there is a line to be drawn between literally describing what someone does and being able to identify how to communicate in English. In this case, the meaning is certainly clear and a Ugandan speaker claims that it is the correct way to speak something in English. Your question narrows this down to Ugandan English which is fair.
Whether we can answer this question revolves around two other questions:
- Is the phrase incorrect in English at all?
- How do we determine if Ugandan English has made an exception to a standard English feature?
In my opinion, the answer to the first question is that “It’s no use of doing something” is incorrect and should be avoided. The grounds for this seem more idiomatic than grammatical. I don't think there is something inherently wrong with the phrase; people just don't use it.
The second question is tougher and the following is the extent of my Google-fu and my personal process of arriving at an answer.
Ugandan English is bizarre and certainly deserves some room with regards to making exceptions. A few examples from Wikipedia (post-formatting) regarding shifted meanings:
"Pop", is used to replace words like bring and come. For example: Danny, pop that bottle here. Heno, pop to my house.
The verb to put on is often substituted for to dress, to be dressed, or to wear. One may hear remarks such as "That lady is rich, don't you see how she is putting on," and "The police are looking for a man putting on a red shirt."
And a good example of some major grammar changes:
Ugandans will frequently combine two sentences into one using the word and. For example, a barber will say "Sit down and I cut your hair," or a messenger might say "They told me to come and you give me the package."
Seeing these examples make me think choosing a different preposition is a relatively minor change and entirely plausible. Some unique behavior regarding prepositions was found via an anecdotal blog (with additional support):
‘to pick someone’ - meaning: to pick someone up. The first time someone told me that they’d pick me from town, I thought: ‘Pick me? Am I an apple?’ I was then told that there is no real time for all those unnecessary prepositions, because everybody understands ‘to pick someone’ just fine.
Actually searching for the phrase "no use of doing" got a few hits referencing Uganda (and a bunch more for India) but nothing I found talked about its acceptance. They just seemed like people talking and could have easily been mistakes.
What is interesting is that I did get a hit on Macmillan:
what’s the use? SPOKEN
used for saying that something is not likely to have a successful result, and so there is no point in doing it
What’s the use of doing something?
What’s the use of complaining?
This is slightly different than "There is no use of doing something" but close enough to again bring the phrase into the realm of "variant" instead of "incorrect."
In the style of Mythbusters, I feel it is safe to call this completely plausible. I was able to find uses of the phrase in English (including one dictionary) and the speaker's instance leads me to tilt toward the answer being yes. Without any better evidence against the phrase being accepted I advise treating the sentence as grammatically correct in Ugandan English.
Edit: The speaker has put the claim in some context. The relevant part:
I said he was right. [...] I can't be sure of anything, I hear it that way and I can't insist that it was right or wrong.
My advice remains unchanged but it certainly seems slightly less apt. I have an email out to the British Council Africa blog.
More research nodes worth picking at include an article in the Cambridge Journal archives I was unable to read and picking up the phone to call a handful of people looking for someone who teaches English in Uganda — an alternative to the phone could be IRC. I don't know anyone in the area. If I cared enough I could probably get a phone trail through the local churches via overseas missionaries.