Hi Beama
Your description of the engine is 100% correct!
So it was made in 1923? that also seems more likely than the 1915 date I'd been given before, all I have to do now is convince the owners.
A Waratah is a Australian assembled motorcycle, with villiers motor and albion two speed gearbox and no clutch, but also no kick starter or pedals either.
Bye the way if the albion box has a number can that be dated? in case the motor was changed some time in the last 85 years, hardly likely but you never know!
Cheers
Barry