80sKid found this mention of the camp in 1938:
The five days journey to Hong Kong was uneventful; the Royal Navy made our journey a very happy experience, and was an excellent host. We were to stay in a tented camp half way up Mount Nicholson, on a plateau looking down on a Catholic monastery, and overlooking the harbour, a beautiful location. In those days, 1938, there were no huge skyscrapers in Hong Kong, no tall buildings at all, so we used to sit up on the hillside, with a full view of the whole of the harbour, a wonderful sight, and especially at night Hong Kong harbour was rated as one of the finest in the world.
And also a photo of the camp in 1939.
The local scout association notes that "The 1st Rover Moot in Hong Kong was held in August 1940 at Mount Nicholson." Rover scouts are older, aged 18-26, and a "moot" is their name for a gathering. It was probably held at this location.
The area was still referred to as "Mount Nicholson Camp" in records of the fighting in 1941.
Based on these I've set the dates it existed as 1938-1941, but if you have any more accurate information, please leave a comment below.
(I've tried a search for 'Nicholson' in HKGRO and the newspaper archives for 1900-1945, but didn't find anything useful.)