nice BSA rifle [M47C-above the M8] , BSA was assigned the serial number range "single digit prefix" 30000-39999 so this is an early one in that series [the second assigned to them]

the rear sight is a second version backsight and yes its correct for the mkI rifle as would be the war expediant flip peep , the marks were designations of changes to both the rifles and the parts or the rifles [complicated but logical] the first version backsight was a milled one adjustable by a fine thread screw phased out early in the war [reinstigated post war] and the second , a formed sheet steel version was implemented to save time ,

the repair to your buttstock looks fine , an arsenal type repair ,