Welcome to milsurps nice looking rifle your correct M 47C = BSA Shirley but it also has a number underneath it been linished out try some chalk on it to see if a snap can be taken I can make out a 2, 0 or O another on the left like a D! under the K.
You may need to give the stock some 50/50 RLO-Mineral turps mix allowing 24 hours between coats until it wont take up anymore as the stock appears a tad dry.
Front top fore wood looks to be a replacement no biggy all part of its patina & journey to date.
What I see where the butt wrist meets the socket is an uneven gap with what appears to be chipping right at the bottom by the socket caused by the recoil on firing so if the wood is dry it shrinks a little allowing a miniscule amount of movement or the rear bolt may not be as tight as it should you need to look into this there is a search bar top right try searching "loose fitting butt." or something like that
Others with far more knowledge will be along shortly to assist you, be aware you never stop at one Lee Enfield.