Rickshaw rank on the western side of Wyndham Street near the junction with Queen's Road Central. The rank had space for 10 - 15 rickshaws. It appears on many photos from the 1940s and 1950s.
Until about 1930, Wyndham Street was "Flower Street" with displays of flowers on the street. Later, possibly with the opening of King's Theatre in 1931, the rickshaw rank opened. King's Theatre closed in 1962, this could be the time of closure for the rickshaw rank, too. This is supported by an article by Fung Chi Ming "The Hong Kong Rickshaw over Time"
....the first of a type of rickshaw fitted with pneumatic tyres which continued in use until the 1960s when the traffic authorities decided not to issue any new licences and to renew existing ones until the man concerned no longer wished to continue his trade. By then, the rickshaw was on the way out overtaken by more modern means of transport.
Addendum: Another rickshaw rank was close by at D'Aguilar Street.