Yes, .38 Webley is basically the same as .38 S&W, not Special. Use lead bullets; they have less friction, and jacketed bullets will be hard to find, anyway. Military .38/200 loads are getting rare, and may not fire reliably, anyway. It's old stuff.
Jacketed bullets (178 grain UK issue stuff) has been known to occasionally stick in the barrel of .38 Smith & Wesson (M&P) revolvers in British service. The bore is slightly tighter than on Webley and Enfield guns, and the wider rifling lands impart more friction. This is a low velocity round, but can be pepped up a bit via handloading. (In some guns; see the current Speer manual.)
If it's in sound shape, just use Winchester or Remington 145 or 146 grain bullets. Because the gun was regulated for 180-200 grain bullets, impact point may be "off." You may need to handload, using cast bulets of the right weight to get best results.
The British jacketed military load has been found to not even penetrate a German greatcoat much of the time. This is not the best defensive round, but will work on small game at close range.
The Webleys are nice guns, although most we see here have dull military finishes. It's an interesting bit of history.
Lone Star