The "Peak Hospital" was founded by Dr. James Cantlie and was the second private hospital built in Hong Kong. Maude Ingall was the first matron, and Sister Annie Thompson, the second, though Dr. Cantlie did not fully approve of the way they ran the hospital. It was originally on Mt. Kellett (listed as such in 1894). It pre-dated Matilda Hospital, by 17 years.
Wikipedia says it opened on 14 Feb, 1872.
M.J.D. Stephens donated the residential house to Canossa, and it was modified and then opened as the first Canossa Hospital
The War Memorial Nursing Home opened on this site on Wednesday, 16th March 1932 [1].
Despite the name, in its early years it was a civilian hospital. The name referred to the fund that commemorated the people who fought in WW1, and which contributed to the costs of building this hospital.
After WW2, the Royal Navy Hospital in Wanchai was considered in too poor condition to be re-used. So the War Memorial Hospital was taken over by the Navy, and became the new Royal Navy Hospital.
Read the history of the hospital here.
Their website has a brief history.
There is an entry in Wiki describing this building, which gives these dates: It was built between 1903 and 1906, and officially opened on July 1, 1907.
Donald C Bowie's 'Captive Surgeon' gives a detailed account of his time at the hospital between 1942 and 1945. During that time he was the Senior Medical Officer, and the hospital cared for Allied POWs.