Martini Cadets were marked Commonwealth of Australia and had the kangaroo stamp on the top of the action where the barrel is fitted.
Your rifle is a BSA sporting rifle ,made for the civilian commercial market .They were chambered in a number of calibers including 310Greener ,32-20 ,300 rook and many more.The take down feature was not used on the military issue rifles.
The lever that is missing is the cocking indicator .It could be easily made from scratch and fitted by any competent gunsmith.