Edinburgh House [1950-c.1980]
This summary of Hongkong Land notes:
Then, in 1950, Hongkong Land completed a HK$7 million redevelopment of 11 and 13 Queen’s Road Central, turning the complex into the nine-story Edinburgh House.
It also notes the demolition:
[The Central Redevelopment] meant the demolition of five old buildings owned by the company—Gloucester Building, Windsor House, Lane Crawford House, and behind them Marina House and Edinburgh House—to make way for the Landmark.
I'm not sure the exact demolition date, but sometime in the late 1970s / early 1980s when the Landmark was built.