There never was a military ".38-200" round
There never was a military ".38-200" round. That designation was a Kynock one for their commercial version of the 200gr lead bulleted "Cartridge, S.A. Ball, Revolver, .380-in Mk I". The Mk I was made obsolescent in 1937 when the Mk II round of the same name was introduced with a 178 gr cupro-nickel jacketed bullet.
.38/200 designation
The British continued to use the .38/200 designation even though they changed to a different 178-grain bullet in 1938 and, during the war, went on to use American-made Smith & Wesson Victory models in .38 S&W as well as the Webley Mk IV .38/200