Elementary adjustable sights were available on black powder guns, such as my S&W and the Colt Flattops mentioned by Phil; and they carried over into the smokeless era. The little S&W single shots are examples of the type.
Colt introduced the Officer's Model Target with screw adjustable sights (windage in the rear, elevation in the front) in 1904 and the New Service Target with similar sights that year or the next. Sounds kind of crude, but they used the type for 35 years and only went to W&E sights on revolvers after WW II; although their target autos had had them for a while.
Blindjustice probably has the first use of screw adjustable sights on a Smith & Wesson in or shortly after 1903. They started with click adjustable sights right before WW II, the S&W .22 Outdoorsman, daddy of the K22, is the best known of the type.
Remington made some Rolling Block target pistols with good sights around 1900.
I don't know about the minor makes. H&R made some pretty decent economy target revolvers with adjustable sights, but they did not come along until after the Colts and Smiths.
Overseas, the 1904 Navy and 1908 "Artillery" Lugers had adjustable sights. I don't know any other continental pistols offhand.
Bisley rules did not allow adjustable sights, so there are few if any early British guns with readily adjustable sights.