The Martini-Henry was a breechloading, single-shot rifle that used a lever-action, as far as I am aware of. I am not well versed in the period at which the rifle was adopted, but I know that it was used in the Zulu wars and I believe the Boer wars. What I am a little confused about is why it became the standard rifle for British troops? I believe that the bolt action was advancing at this time and effect rifles were made from that, and there were other mechanisms too. As far as I am aware of, the reason lever actions were not fully adopted by countries in previous wars, was that it made it hard to reload while prone. Was that really a big issue? Or were there aspects of the Martini-Henry that made it so effective.
Basically, it was much better than the Snider conversion of the Enfield muzzle-loading rifle it was replacing. The Snider took more steps to reload -for the Snider, the hammer had to be cocked and the breechblock pulled to extract the cartridge. The Martini-Henry ( or what should be called the Peabody-Martini) eliminated those steps. The older Sniders also had the annoying habit of blowing open , right there by the shooter's face, when there was a cartridge rupture...and in the early days of the development of cartridge cases, that was not unusual. The Peabody-Martini action was also quite strong, easy to take apart and clean, and the rifle was lighter and the new 450-577 cartridge had a longer range than the 577 Snider.
Although there were bolt action designs being made, it was early days for them. The US had looked at the Ward-Burton in 1866 and had rejected it ( wisely) with many others ( not so wisely) in favor of their own rifled musket conversion. The French and German armies had bolt-action designs in the later 1860's that didn't have metallic cartridges and so had problems.
Mauser would bring out the first model in 1871, right when the Peabody-Martini came out. The Swiss had noticed the Turks' success using Model 66 Winchester rifles against the Russians and had developed a bolt-action magazine rifle, the Vetterli, at about the same time. If the British had been able to see into the future, they might have come up with their own bolt-action magazine rifle. But they couldn't, so they just settled for something that was a lot better than what they had. In the 1870's there would be plenty of others doing the same: the Belgians, Bavarians, Austrians...and the US. Generally speaking, there would be a European arms race for the rest of the century, and armies would not add improvements to their arms because of theoretical possibilities but because somebody else had gotten a a little ahead in the race. The French adopted the Chassepot because of the German Dreyse, the Germans adopted the Mauser 1871 because of the Chassepot, because of the Mauser the French reworked the Chassepot to be the Gras, the Germans in response added a magazine to the Mauser, the French created the Lebel that had both magazine and smokeless ammunition, and the British would try to catch up with the Lee-Enfield. In that race, the bolt-action simply won.