I am envious - having trouble locating this level of No.4 here in Texas. Patience I guess...
Others more expert can comment, but very interested in the cocking piece. Yours looks just like the First Variation on my SMLE Mk III* from 1916. Stratton shows a First Variation for the No. 4, but it is dished and round. My SMLE is 'D' shaped, flat, and knurled around the perimeter, like yours.
If Long Branch didn't start production until 1940, what were they doing with an early 1900's-era SMLE part? Did the Brits ship over stock?