When you say snow, do you mean a) snow high on the mountains or b) snow on the ground you can play in?
If a), Snow normally starts falling on the peaks from late April, however the sun is still pretty warm at this time so it melts in a few days. Snow falling June-August can last for weeks in the shady parts of the mountains, however it can be erratic, if there arent any cold fronts there can be no snow at all. Last chance for snow is normally around October in a normal year.
If b) You really cant plan this type of thing in advance. Snow might only fall at "resort level" once or twice a year at best, and unless its a very heavy snowfall, it might not even last a day or two. Note that the Southern Berg has a much higher incidence of ground snow due to higher elevation and being further South. The heaviest snow usually falls between End July and Mid September.
Last year there was extremely heavy snow in August, and it managed to last about a week even at lower elevations.
I took this pic from just outside Drakensberg Gardens Hotel
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