I would say that there is little doubt that what you have is one of the many S&W revolvers acquired by Britain during WWII (officially designated "Smith & Wesson Pistol No. 2") for issue as "substitute standard" in lieu of the official British handgun at that time, namely the Enfield "Pistol, Revolver, No. 2", chambered for the British service cartridge originally referred to as the .380-200.
In fact, that cartridge was nothing more than a British military variant of the .38 S&W cartridge, initially produced with a 200 grain lead bullet until it was decided that contravened the Hague convention against expanding bullets in war, after which a 178 to 180 grain jacketed bullet was used. (
http://en.wikipedia.org/wiki/.38/200)
Britain
did acquire .S&W revolvers during WWI - but they were "Hand Ejector" revolvers built on the larger "N" frame and chambered in .455, the standard British service revolver cartridge then.
The story behind Britain first acquiring these .38 revolvers is interesting ..... In 1939, S&W agreed to design (and later manufacture) what became known as the "Model 1940 Light Rifle, 9mm" but the design failed service trials. However, S&W were unable to repay the money advanced by Britain, having prematurely tooled up for production. They were saved from bankruptcy by acceptance of their offer to work off the debt by supplying the British with service-suitable revolvers. This S&W was able to do quite easily since .38 S&W was one of the standard chamberings of the "Military & Police" series of their commercial ".38 Hand Ejector Model of 1905, Fourth Change" built on the "K" sized frame. (Note that these were accordingly not "Lend-Lease" arms, they were effectively bought and paid for. Mind you, later on some revolvers were provided under the Lend-Lease program ....)
It is not well-known that this model of .38 S&W revolver was Canada's
primary official handgun during WWII (adopted quite independently from Britain) - we acquired in excess of 118,000 of them!
To clarify further: the "K" frame "Military & Police" revolver is precisely what a U.S. "Victory Model" also is - albeit chambered in .38 Special. As the war progressed, the quality of finish on all of the military-issue revolvers (i.e. British, Canadian and U.S) was progressively downgraded.
FWIW, here is an image and caption scanned from Chamberlain & Taylerson's
"Revolvers of the British Services, 1854-1954" -