Oh Jambo, I'm in full "lust" mode over here.
From my experience in both B.C. and Ontario, when no title papers are ever found you can swear an affidavit at a recognised agency like a lawyer's office authorized a notary public, etc., and work through the licensing bureau to ascertain if the vehicle is lien free by searching serial numbers ....et voila, new title issued. I am sure Quebec has similar procedure available.
Oh boy, what a treasure you've stumbled upon. Those were considered premier motorcycles in their day along with Douglas, Velocette, Royal Enfield, Norton, Ariels and Vincents. BSA's and Triumphs were slightly more pedestrian in the pecking order until most of those others withered due to higher production costs. Norton hung in there for a while but it and BSA couldn't find enough interest to compete with the Japanese machines.
Triumph has re-entered in the last decade but the cachet of a new model Triumph is far different than it's predecessors and why bother when you could buy a rompin, stompin Ducati that will run the jock-strap off any modern Triumph.
These guys would love to hear from you and have a wealth of experience with your current dilemna.
https://www.cvmg.ca/