My grandfather, George Lawrie (1909-1965) served in the Military Police during the Second World War; spending the duration mostly in North Africa. I recall from a very early age pictures of him in the desert on an old motorcycle. Some old photo albums of my Gran recently resurfaced and a photo of him astride an M20 was amongst them. Great Stuff!
It just so happens that there is similar early 40's military spec M20 up for auction at nearby Glenview IL; part of a lifetimes collection by a chap called Lee Roy Hartung of automobile's and motorcycle's. Hartung specialized in interwar items including: Hendersons, Excelsiors, Harleys, Indians, Flying Merkel and a 1913 Pope. Outstanding selection of important bikes from the teens, twenties and thirties. I spied the BSA amongst them on the aution page; here are a few detail shots of it in splendid original patina.
Given the coffers, and space, I would try and get it running as-is, and enjoy its simple thumping ride as my grandfather would have.