I don't think a Lithgow, friend; generally they marked theirs rather well.
I would think more likely a BSA built under the Dispersal program. At that time, BSA had the only plant in the UK capable of turning out the SMLE. They made some nicely-marked ones in 1940, then the Luftwaffe decided to visit the factory..... which they did quite impressively. Following that, production was dispersed, with parts being made in dozens of small factories and brought together only for assembly. At the same time, the NAME of the company was removed from the rifles and CODES introduced in the hopes that the Luftwaffe might forget where Birmingham was.
A Lithgow, even in the worst of times, would have been marked at least "MA" above everything else.
Photos would really help in this case.