Actually (yes, I said it!) the .38/200 was designed in the 1920s - and NOT by Webley!
Webley had used the .455 round in its military pistols for WW1, and the British Army found that the common "never touched a gun before" conscript had problems dealing with the recoil of that round, so in the early 1920s they looked for a less-powerful round (NOT "in the black powder era").
They settled on a modified .38 S&W round with a 200 grain lead bullet - hence the designation. Webley had been approached to built a version of its standard military pistol for the round, with the gun a little smaller and with some changes to make it easier to make and use - but the pistol Webley submitted was only smaller, they had not incorporated any of the requested (required) design changes.
Therefore, the British government had the government-owned Enfield factory make a near-copy of the Webley with the desired changes, which entered production and service as the Enfield No. 2 Mk 1 pistol. This did cause Webley to sue the government, but they only received a symbolic victory, and only after large numbers of the Enfields had been supplied to the government. Webley did build many of their Mk IV revolver (the same one the government had rejected) during WW2 - about half as many as the total of Enfield No. 2 Mk. 1s.
The round was changed in the 1930s to a 178 grain jacketed round to satisfy the lawyers' concerns that the lead bullets could be considered "expanding/exploding" and therefore violate international treaties.
The standard modern .38 S&W ammo is perfectly safe to fire in any Enfield No. 2 Mk. 1, Webley Mk IV, or other revolver chambered in .38/200 that is in good mechanical condition.