These initials were probably for William & Charles Scott, well known gunmakers on Bath St. and Bagot St. in Birmingham during the P53 period. This firm merged with P. Webley & Son to form the famous revolver manufacturer Webley & Scott (1897).
I have seen several P53s made by this firm, and the initials W&CS often appear on the bottom of the barrel and/or the back of the lockplate. I have not previously heard of a bayonet with this marking.
Hope this helps,
Geoff Walden