The joys of the (genuinely) vintage motorcycle!
I have a collection of unusual motorcycles for which there is no expert to consult. Information is usually hard-won, but the satisfaction level is very high as each snippet falls into place to build a picture of both the bike to be rebuilt and the people who put it together.
Cotton shouldn't be too hard. They were present at most of the major shows, they exported and they advertised. They feature in books and there are numerous survivors, and the suvivors have owners some of whom are sure to be knowledgeable.
As a starting point, can I suggest that you bolt together what you have, and take a couple of decent photos so we can see what you've got.
On the research front, collect every photo of a vintage Cotton you can find and order them by age.
Cotton models changed over the years, so from the details of the parts and the frame number I'd say there'd be no problem coming up with a date and model.
Leon