I've answered a very similar question here. I'll put my answer here as well, as it seems to fit here with a few extra bits.

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Lever actions were used in WW1. The Royal Navy and Royal Flying Corps would procure several tens of thousands of Winchester Model 1886 , 1892, and 1894 rifles in their original chamberings, and the Russians, ever desperate for arms, placed several large orders for Winchester 1895 rifles modified to mount bayonets, use stripper clips, and chamber the Russian 7.62x54r cartridge for a total of nearly 300,000 examples delivered by the time Russia left the war.

I can't speak for every nation's individual rifle trials, but it helps to look at the history of lever-action rifles to understand their limitations. The Winchester lever-action rifles of the 1870s and 1880s were very capable designs, but they were limited by the strength of their action. The Model 1873 and 1876, for example, couldn't handle the power of the .45-70 cartridge that was the standard of the US Army at the time, and we don't see lever-action rifles capable of chambering standard military rifle cartridges of the day until the development of the Winchester 1895 and Savage Model 1892. That alone presents a major problem towards the adoption of lever-action rifles as a standard infantry arm. Adding to that were a variety of doctrinal concerns that varied depending on when specifically we look. Prior to the advent of smokeless powder, magazine rifles were seen almost as a liability more than an asset over logistical concerns. Although magazine rifles had existed and seen combat since at least the US Civil War, militaries across the world were generally very reluctant to consider such systems over fear of undisciplined soldiers firing off all of their ammunition in a panic and creating problems with supply of ammunition. Even as magazine rifles finally caught on in the 1880s, we see this thinking persist through WW1 in the form of the magazine cutoff, and it would play a major part in the selection of service rifles for several major powers including the US and Britain. Later on, once magazine systems had finally been accepted, the specific type of magazine in most lever-action designs was a problem. Although tube magazines did make a showing on several military rifle designs like the Kropatschek, Gewehr 71/84, and 1886 Lebel, most nations were averse to the concept. Originally, safety concerns dominated - the British specifically would eliminate tube-magazine rifles from consideration after one such rifle suffered a detonation in the magazine caused by bullets resting on primers, and several other nations independently came to similar conclusions during their rifle trials. Although such concerns aren't necessarily what they were made out to be once ammunition quality improved (note that the Lebel never gained a reputation for safety problems relating to the magazine), the advent of charger and en-bloc clips to speed loading created another major concern for tube magazines. Unlike vertical box magazines, tube magazines have to be single-loaded. In practice, that meant that charger- or en-bloc-clip fed rifles would have a significantly better sustained rate of fire compared to a single-loaded tube-magazine rifle.

Most of the rifles used in WW1 were developed rapidly in the aftermath of the invention of smokeless gunpowder - France had the Lebel in 1886 and Berthier in 1890, Germany adopted the Gewehr 1888 in that same year, Austria adopted a Mannlicher straight-pull design, Britain adopted the Lee-Metford, and Italy and Russia adopted their modern rifles (the Carcano and Mosin-Nagant, respectively) in 1891. The rush to adopt a modern smokeless powder rifle meant that nearly every major power had adopted a new rifle before a viable lever-action had even materialized. We do see some examples of lever-action rifles making it to various trials during this period, but without success. The British would look at two lever-action systems in the early stages of their Lee-Metford trials (of a total of 40 designs submitted), but none made it to any comparative trials. The Savage Model 1892 was submitted to the US Army during its smokeless powder rifle trials, but the US Army opted to go for the Krag-Jorgensen rifle in large part thanks to the Krag's unique magazine system. The New York State National Guard would hold trials in 1895 for a new rifle in which both Savage and Winchester would submit lever-action designs, and ultimately Savage would come out as the winner, but no orders ever materialized, as New York ended up receiving surplus Trapdoor Springfields for free.

The Russian contract for the Winchester 1895 really marked the only example of lever-action rifles being used by frontline troops. It was ordered largely out of desperation due to a serious shortage of rifles, but compared to the other stopgaps Russia procured (including several models of black-powder cartridge rifles and the modern Japanese Arisaka Type 30 rifle), the 1895 actually worked out reasonably well. The design was easily adopted to 7.62x54r and only required very simple modifications to use the same charger clips that the Mosin-Nagant used. Minor modifications were made to make the rifle better suited to military use, including an upper handguard and bayonet lug (complete with bayonets unique to the rifle), and the sights were calibrated to the same ranges and with the same units as Imperial Russian Mosin-Nagants. Unfortunately there's not much out there detailing their use in combat, but Winchester 1895s were given to the Latvian Battalions (among others), and they would serve with these units through WW1 and into the Russian Civil War. After the Russian Civil War, they were put into storage along with other non-standard rifles, and many examples were supplied to Republican Spain during the Spanish Civil War. The only major concern with the design was how working the lever left the bottom of the action very open as it was being worked, but there doesn't seem to be even anecdotal evidence claiming that this created any practical problems, and extant examples of Winchester 1895s appear to be well-used, indicating that reliability issues weren't common enough to keep the guns out of combat.

One thing to note relating to your post, however, is that lever-action rifles aren't necessarily faster actions than a bolt-action. While speed very much depends on the specific actions in question, several bolt systems like the Lee-Enfield have a reputation for speed, and the Winchester 1895 in particular had a unique lever system that made the action a bit slower than many other lever-actions. The larger cartridges the 1895 was designed for required a significantly different mechanism that pivoted the lever further from the action and about a different point on the rifle, leading to a clunkier action than one might expect.