It has been taken onto the books in New Zealandduring WW2. (the NZ and rack number stamped)
The "B" on Savage made rifle bodies is quite common; I do not believe that it has much to do with BSA. The "B" on the rearsight could indicate BSA though.
The shaving of wood from the rear of the fore-end is to fit the mounting plate of competition target sights - adjustable iron sights rather than a scope.
The bolthead marks seem to be the Savage "square S" and proof/acceptance marks.