A common thread running through the early motorcycle manufactures was their ability to produce small precision components, whether it be cycle parts, textile machines or, in the case of BSA, guns.
The early years of what would become BSA motorcycles were dedicated to the mass production of small arms (rifles) for the British military.
The ability to produce armament in ever increasing quantities required the ability of the manufacturers to both mass produce components and also to make parts interchangeable (in years previous, skilled fabricators had produced one-off items).
For BSA the ability to mass produce precision components for guns was easily transferable to producing firstly cycle parts, then complete motorcycles.
The early history of BSA is one of a company looking for markets to fill with their precision engineering capabilities.
Further reading:

Besides ‘Birmingham Small Arms, it was also heard once in a while to stand for ‘Bastard Stopped Again’.
They did however have some GREAT ads back in the day…
You recall the Lightning half out of the crate, and the drop dead gorgeous girl half out of what few clothes she had on? !