Unlike other museums, the Australian War Memorial has chosen to make the original vehicle registration books accessible to the public.  All original books have been scanned page by page and put into large PDF files for everybody to search. The books give some history for each vehicle.
 
The records show Australian  Imperial Forces (AIF) and Australian Military Forces (AMF) vehicles of all sorts.
(Difference between AIF and AMF are shown on the AWM website pages https://www.awm.gov.au/atwar/structure/one-army/ and https://www.awm.gov.au/atwar/structure/army/.)

The books do not contain  the motorcycles used by the Australian army in the Western Desert (British Army stock) and will therefore not give all answers.
It will also not give information on MC's used by RANavy and RAAF.

To ease the search for a specific engine number I have categorised them all in one excel file.
If you want a quick answer, drop me a line and I can tell what C number(s) fits to your particular motorcycle if in the list.
 wd16h@telfort.nl
  
It appears that a single motorcycle can have had up to 4 different C numbers over its life, caused by transfers between the AIF ad AMF.
With the census numbers it is much easier to go through the various books.

All the books themselfs can be viewed and downloaded via the links given here. I only added the ones which refer to Nortons.