Hi Matt,
If you've got all the parts in the photo, you have a very interesting, and very restorable, motorcycle.
This type of machine is pretty typical of the era. The mostly likely scenario is that Watts would have bought in most of the parts - motor, gearbox, forks, saddle, frame lugs, wheel parts, etc. - and assembled the bike themselves. Probably a bit less likely is that they bought the complete machine from a manufacturer with their Watney brand painted on the tank. Either way, it's pretty sure that few were made.
I have two suggestions: join the Vintage MCC, and find someone local who is familiar with restoring vintage bikes and who is prepared to help you out. The VMCC has a Blackburne Marque Specialist who can provide info about the motor. The Brampton Bi-flex fork, the medium weight Burman gearbox, Brooks saddle, probably British Hub Company hubs etc. are all pretty standard, so the restoration mostly depends on the level of wear and decay.
Don't be in a hurry to pull everything to bits, don't sandblast everything, and avoid powder coating and chrome plating.
Do your research first: found out whatever you can about Watney motorcycles, and also about the bits it is made from.
Plan your restoration: even if you're not going to do it all yourself get a copy of "The Vintage Motorcyclist's Workshop" by Radco.
Above all, enjoy!
Cheers
Leon