The .315 that was chambered in the Lees was actually what was called, at least in England & the Empire, the .500/320, but which was nothing other than the 8x50R Austrian Mannlicher.
It is actually much rarer over here than the .375 Fl. N.E. 2 1/2", although not as rare as a Lee Speed chambered for the 7x57 Mauser.
The .32-40 chambered rifles were not actually Lee Speeds & the .256 Fraser Flanged [6.5x53R] complete rifles were never made by B.S.A.Co. or L.S.A.Co.Ld., the only 2 manufacturers of Lee Speed & the later commercial Lee Enfields like the OP's rifle, but were made up on B.S.A.Co. commercial actions by Fraser, McDougall & a couple of other Scottish "makers" for highland stag, as cheaper alternatives than Holland's M1895 Dutch Mannlicher or Farquharson-based single shot rifles in that chambering & to compete with Jeffery's offerings, also based on the same actions.