The British Army, looking for a more modern action to replace the Snider, eventually settled on another American design. The Peabody. Henry Peabody's gun would have been the new rifle of the U.S. Army if not for 2 things. The end of the American Civil War, and the huge stocks of Springfield muskets that were laying around. Cost ruled, as in the case of the Snider, and the Springfield Trapdoor was born.
Image from Adam's Guns
The action that caught the Brits attention was the Peabody-Martini. The Swill born Martini had modified Peabody's gun by adding a striker in place of the external hammer. The gun eventually adopted was a Martini-Henry, which used Alexander Henry's rifling system. The early guns had trigger and extraction problems, but were solved in later marks and with better ammunition. It could be said that the resulting gun is one of the finest and strongest breechloaders ever made.
The bullet for this new gun was the .450/.577. It used the same size base as the Snider, but the construction led to problems. Originally an iron base with a boxer primer, with coiled thin sheets of brass soldered to it, it was susceptable to damage. The cartridge was finally made from drawn brass (like modern bullets) which solved the extraction and loading problems.
The Martini-Henry had basically the same rate of fire and ballistics as the Snider, but had some advantages. Once the ammunition and the extraction was worked out, the Martini had a huge advantage in reloading, as the spent case was thrown clear, and a new round could be inserted immediately. In the Sniders case, the action had to be opened and the breechblock drawn back, and the rifle turned upside down and shook to release the shell. The other was that the action was inevitably stronger, due to being purposely designed as a breechloader, as the eventual rebarreling to .303 proves. The later British .303 round pressures were more than double the .450/.577.
The Martini-Henry stood up well right away. In 1879, At Rourke's drift during the Anglo-Zulu war, 150 British troopers expended close to 20,000 rounds in defense of their mission. The earlier defeat at Isandlwana seems to focus on ammunition shortages, not any problems with the rifles. The rifle was in service for 30 years and some units still used them into WW1. Later variants included the Martini-Metford, which was an attempt to stop barrel corrosion from Cordite powder, and the later Martini-Enfield, which used the later British .303 round.
Martini actions are still being used by custom gunsmiths the world over for Shutzen style guns and others. You can still get original guns at Atlanta Cutlery, and they show up on Gunbroker amongst others. Reloading the .450/.577 is expensive but can be done. (not the coiled cases, though.) The original guns kick like a mule, but Martini actions have been converted to any bullet imaginable. A truly interesting weapon!