Ok people, I've tied this up long enough.
It's not in the Blue Mountains, just that they they had an article in their newsletter about the builder of this engine.
The point about the gauge of the riding trolley is that it not the same as that of the engine. The engine is a 4-8-8-4 "Big-boy" Mallet design. It is 'O' gauge but was still able to pull a couple of people. In the photo, taken in around 1961, it was almost completed and not yet painted.
This track gave me my first opportunity to learn to drive a live steam loco. The builder also introduced us to the track at Darvall Park. At that time the main track at West Ryde was elevated with 2 1/2" and 3 1/2" gauges, and a little new 5" gauge track on the ground which seemed to be a novelty.
Like LBSC, the builder of the loco was English and a contributor to ME magazine. The backyard construction could easily have been in a London suburb, but it was in a Sydney suburb. The builder went on to distinguish himself by designing and building ever smaller engines, down to N or maybe Z gauge live steam and 1/8" gauge electric (that's 1/480 scale).
The riders in that photo both had fathers working in the Mechanical Engineering Department at Sydney University. As far as I know neither of them grew up to be a Queensland politician.
The Nottingham mention was just a cryptic clue.
This competition depended on knowledge of the engine, if so then the answer is obvious, otherwise impossible.
So where was Sydney's only man-carrying O-gauge line at that time?Mr. Arthur Sherwood's backyard in Chatswood.
(I have not done him justice here - Google him for stories about his contribution to conventional model engineering, ultra-miniature model railways and medical technology.)
I win this time, but as I don't have another photo ready to post it's open floor until I find another!