Member "boltaction" above nailed it with his assessment. I don't have much to add to what he said (it is a commercially produced BSA rifle, intended for sale to civilians, not a conversion from a military rifle), except to say that it was not a conversion from anything---it was a never a .303. It was built from the start as a .22. The BSA company advertised such rifles heavily in the years leading up to WWI. I will include a section on them in the Lee-Speed book; there were several variations. (Promo, please look for a private message from me. I would be grateful if you would let me use a photo of this rifle in the book.)
One bit I can add is that Clabrough & Johnstone had nothing to do with this one. I have studied their conversion work and this isn't one of theirs. (There is an excellent study, BTW, on the Clabrough & Golcher/Clabrough & Johnstone companies, titled "J.P. Clabrough - Birmingham Gunmaker" by Lawrence Shelton, published in 2013. It was reviewed in the March/April 2015 issue of "Shooting Sportsman.” The story is interesting: a Yorkshire gunsmith takes ship for America around 1860, crosses the country in a covered wagon and ends up in San Francisco, where he builds up his own gun business. Educating himself on the market for good English shotguns in the American West, he returns to England (Birmingham this time) to open a factory with his brothers to build guns for the American market, and sell them in his San Francisco shop, which he does successfully (percussion, hammer, and hammerless models in turn) until the McKinley Tariff stops his operation in its tracks. Shelton details all the models and variations, production figures, etc., and tries to evoke the era. It also covers the British Bull Dog pistols that Clabrough imported in large numbers. Lots of good stuff there about shotguns, the British Gun Trade, the market for shotguns among shooters and hunters in the American West. There is not much Lee-Enfield-related, but he notes the .22 Lee and Martini SMRC conversions, and identifies some markings to be found on them.
If you look for the book, be careful not to mistake it for an earlier (very different) book on Clabrough with a similar title that Shelton did back in the 70s. This one is a totally new book, not a second edition.