Webley-Fosbury automatic revolver is never developed and the money and man hours are instead dedicated to refining the more conventional automatic pistol, which the Army adopts before WWI alongside the Navy.
Webley-Fosbury was, like most Webley products, developed on Webley's dime and then sold privately to military officers, so not much money came from the Government to start it. If fact, in this case Major Fosbury would have shouldered some of the cost himself, which would have reduced the weight on Webley's shoulders as well. What's more, it gave Webley, and the man who developed their automatic pistols, experience with the concept, as well as some profit that came in handy when gun regulations were increased in the early 1900's. If you want to bring about a faster British Automatic Pistol I would say you could go two ways:
1) have the Webley-Fosbury come onto the market sooner so that a respectable amount are sold to officers heading for the Boer war. This helps Webley develop the design of the Webley 1904 earlier, realize the problems with it and get something closer to the 1906/1907 or even 1910 model in time for the 1904 trials for a cavalry pistol.
2)Either instead of or as well as 1), prevent the rise of General French and Haig to quite the position of prominence that they reached OTL, preferably by bring about the victory of the "firepower school" over the "morale school" of officers as opposed to the opposite which happened IOTL. Increased interest in mounted infantry tactics for the cavalry and the firepower of the soldiers in general is likely to expand the interest in obtaining an automatic pistol for the Army. A niche which Webley's design would probably be the best fit for between its refinement in 1905/1906 and at least the advent of the Colt 1911. Even then Webley's familiarity with the British Military and British origin would be advantages over Colt.