Okay, here's my best shot at an interpretation of this 22RF trainer.
This appears to be a No2 MkIV* (where the * indicates it has the empty magazine shell catcher), and the markings on the bolt head are consistent with this pattern (model) designation. It was converted from a 1919 SMLE (No1 MkIII*) during WWII either at Parker Hale, or at RSAF (Royal Small Arms Factory) Enfield using a Parker Hale lined barrel, and I lean toward the latter. An interwar No2 MkIV* would have the pattern information stamped on the right side of the butt socket but that was not always (ever?) the case for the WWII conversions. The EFD 44 is an Enfield view (inspection) mark where the "44" is likely just the inspector's assigned number that is coincidentally one of the WWII years, but I'm not certain about that.
The barrel was lined and rifled by Parker Hale (if I'm correctly seeing a concentric ring indicating the liner at the muzzle) and attached to the receiver (different serial numbers on the receiver and barrel). Both receiver and barrel have the appropriate King George (Georgius Rex, GR) royal cypher over P proof marks from when it was proofed originally for the .303 MkVII* (HV marking) cartridge. Under the proof mark on the barrel is an LSA (indicated by the X) view mark, so the .303 barrel was originally made at London Small Arms Co. (LSA) and later relined and fitted to an SMLE receiver made at Enfield. The additional view mark of broad arrow over EFD (Enfield) over 7 to the left of the original proof mark on the barrel further suggests that this was assembled and/or inspected at RSAF Enfield when it was converted to the .22lr trainer.
The "BV BP NP .22LR" is Birmingham Proof house, Nitro Proof when it was sold out of service and proofed for commercial sale (not sure what the BV stands for).
Thanks to Ian Skennerton's books for what I know but all errors are mine.