Welcome !
From your use of the words "proper" and "odds", as well as the question about unleaded gas (leaded's been unobtainable in the US for the road for 20 years), I assume you're in the UK.
Well, since you don't want to be hard on a poor widow but don't want to be into this bike for too much, I'd suggest £1400 as a starting point. She may be glad to get it, OR her nephew (who lusts after it and is hoping to get it from her for love) may have told her that it's a rare creation, it's like a Vincent or a Brough, this BSA, and it's worth £12000 as is, better hang on to it till I can fix it up for you ...)
You don't need to do anything to convert it to "unleaded". That sort of turned out to be the Y2K of motorcycles - lots of worry but nothing really happening.
Availability of running parts, bearings, engine parts, etc is good, easier than japanese bikes. Tinware can be expensive, especially short-run special tanks and nacelles. If the bike doesn't have good tinware as is, you best knock the price down low or avoid it.
Take the bike apart and do it back up as a dry assembly before having anything painted or chromed or powdercoated. You don't want to find out too late that the part you just spent £200 having chromed is missing a bracket that now needs to be welded on. A lot can happen in 55 years, there's NO guarantee that anything's stock.
Join the BSA club; lots of good help and encouragement there for very little money. Get a good set of factory parts and service manuals, you can't do without them. There's lots of expertise out there but no one wants to be used as a walking talking service manual.
There's more - I'm sure others will be along!
Lannis