Hello, & welcome. It's difficult to say too much about your rifle from the limited views, but it appears to be a genuine BSA Shirley No4 Mk1 (T) of 1944 manufacture. It would likely have been blued or brunofixed originally, so has probably been refinished with black suncorite paint (looking at a blown up view of the front pad). It is quite possibly a 'Less Telescope Rifle'. These are not uncommon, & are rifles where Holland & Holland effected the sniper conversion except for the fitting of a scope & bracket. Many of these rifles were subsequently fitted with scopes further down the line, & of course some have had this done in recent years by civilian owners. This may be why there is no receiver side wall T & why the scope number looks a little 'fresh'.
More photo's would probably generate more information for you.