Looking at your pics is it defiantly a P14? ie a 303british? eg there is a Remington model 1934. Also I think Winchester did a commercial action? For a serial number I'd expect to see a Parker Hale number on the receiver. The Wxxxxx number is heavily stamped on the rh side near the Knox and on the bolt. Parkerhale tended to grind off the military numbers (on No4s) and add their own mind you.

If it was a BSA i would expect to see the 3 rifle logo. A Pprkerhale and I'd expect to see parkerhale Birmingham on the knox.