I, personally, would think they'd be more popular than bolt-action rifles, since they'd probably be able to be fired faster.
There have been a few examples of lever action rifles being adopted by militaries, such as usage seen during the American civil war and various Indian conflicts of lever rifles such as the Henry rifle, among others. However, the lever action possesses several notable shortcomings when compared to a bolt action:
-Awkwardness of shooting from prone or rested: The need to have a clear space underneath the gun can make cycling a lever action awkward compared to a bolt action from a prone position (lying down) or rested on a surface, such as a trench parapet. Obviously a lot of shooting by soldiers is expected to be done from these two positions.
-Reliability: The lever action is more vulnerable to fouling and jamming from dirt due to it being located underneath the gun, especially when shooting from prone. A bolt action is inherently less vulnerable to this problem.
-Ease of reloading: From a kneeling or standing position, a lever gun can indeed be cycled faster. But with a tube magazine, reloading the weapon generally involves feeding the tube individually with rounds, which is time consuming and awkward in battle. A bolt action design on the other hand lends itself far better to being loaded using stripper clips, which allows for greater overall rate of fire compared to a levergun.
-Less suitability for modern rifle cartridges: When the lever action appeared on the scene, the rifle round of the time was generally composed of a relatively short (compared to later cartridge designs) cartridge case, with a round-nosed solid lead bullet propelled by black powder, similar to pistol cartridges of the time. These rounds generally produced lower chamber pressures which lever actions could easily handle, and the rounded, shorter shape of the cartridge lent itself well to the lever action/tube magazine design. However, rifle rounds began to modernize rapidly; modern ammunition made use of longer cases with high pressure smokeless powders and pointed, not rounded, jacketed spitzer bullets, as designers sought to increase the muzzle velocity and range of infantry rifles. This did not lend itself so well to lever actions, as they were less able to deal with the pointed long bullets and their much higher chamber pressures. Bolt action designs were far more suited to cycling these new designs, and were generally more structurally robust, and thus better able to deal with the higher chamber pressures produced.