Hi William,
I own a half a dozen BSA bikes, the earliest being an Oppenheimer Special from the mid thirties. From your photos, the only part that I can say for sure is BSA is your crankset. BSA sold many parts to other builders, so it is common to see BSA parts on other bikes. They even sold lugs to other builders. Does it have a BSA head badge? Is the coaster brake an Eadie? It looks like the top tube tube and down tube is bent a few inches behind the head tube. You can view many BSA catalogs here
http://www.veterancycleclublibrary.org.uk/library/