eurasian_david found this report of a well at 4 Queen's Road Central being closed in 1899:
“A WELL TO BE CLOSED.
Mr. F. Browne, Government Analyst, certified that the water obtained from a well at No. 4 Queen’s Road Central was so tainted with impurities as to be unfit for potable purposes and likely to prove injurious to health.”
I think I've read of one of the big banks in Central using its well for water during the fighting in 1941. There's also a 24 Dec 1941 diary entry from Henry Ching talking about wells where he lived in Happy Valley:
We had to do something about water. Among its preparations, the Government had made ready concrete slabs to be fitted into the nullahs which perpetually carry mountain spring water to the sea, thus to form little local reservoirs. One deep nullah runs past our front door. The slabs are there, but no one put them in position. The water anyway is not usable. The nullah is carpeted with garbage, and on this now rests a corpse. We explored our own flush water well, but oil from the pump had fouled it badly. Our neighbour was more fortunate; the water in his well was clear and sweet, and we drew from there.
The Hilton Hotel was using its own well for water during the 1963 rationing (see https://gwulo.com/atom/30352), though they don't say if the well was on the hotel site, or elsewhere.