I'm fairly confident that double decker buses were used on (what is now) the No. 96 tram route from the late 1930's in the years between the last cable tram and the first electric tram. They were supposed to be a short term measure, but track relaying and installation of the wires was delayed by the war and I think the double decker buses ran on that route for almost a decade.
And of course, double decker trams famously ran in Hobart.
You are correct, double-deck buses ran on the Bourke Street cable tram route, though your dates are slightly wrong. The cable trams definitely worked in to 1940, may have been around October 1940 they closed. The reason they closed then was that the equipment was worn out. The double-deckers lasted a little more than a decade. I think the electric trams were introduced on that route in 1955 or possibly 1956, but the double-deckers had been taken out of service a year or two before then. They had a problem, and that was they tended to have collisions with the verandas on the shops!
I remember riding on the top deck of a double-decker in 1952 with my older brother. We were visiting the All Models Exhibition at the Exhibition Building. A wonderful exhibition of aeroplanes, boats, trains and road vehicles. I was only seven years old at the time, but still remember it well, and still have the exhibition booklet which was published for the event.
With the double deck trams in Hobart I believe the reason the upper deck was open was to improve stability in high winds. Having said that, on all the visits I have made to that beautiful city the weather has been extremely agreeable. And they average more hours of sunshine per day than Melbourne.