Great photo of your ancestor there !
The bike is going to be a BSA, as you say, and circa 1920s
It has drum brakes, and a flat-ish tank, still under the top tube of the frame.
Motor is a sidevalve (not ohv) and the magneto is out the front, behind a somewhat pointy looking cover, you can see bits of oil pump on that cover. And the muffler has a fishtail.
No lights - lights were optional back then, and squeeze bulb horn (so no electrics).
After dark could have been a bit of a problem - and not allowed.
Now, BSA made quite a range of models - something for about everybody, but that is leading to what is probably going to be something like this 1927 Model B, a neat 250cc Model B or Model B deluxe. BSA were a very large manufacturer, and probably had a local agent to be able supply this, a well respected maker. Someone who knows their BSA's could probably pin it down more closely - this was the famous BSA 'round tank' model, after the tank had become wedge shaped. (?)
This is not yet a perfect restoration, but is getting there - your pic could give some pointers
A sprinkling of most BSA years and models appears for sale now and then, you could probably source one close to this spec off fleabay or out of the classic magazines with some diligent searching. Being a flat tank (vintage) bike means (maybe) a bit of a premium over say a 1930s or 1940s model, but well affordable. Joining your local Vintage Club, or enquiring of them may well produce some leads too, and you will meet other bike enthusiasts. Most early bikes will need regular fettling, they were not just ride and forget machinery back then.
hopethishelps.
https://www.bonhams.com/auctions/15320/lot/457/http://motorcycles-for-sale.biz/sale.php?id=53310
