At a glance, it does seem really strange that autoloading infantry rifles took so long to become the standard for every army, but when you really look at things, you'll see that armies generally recognized the utility of an autoloading infantry rifle, but were generally stymied by a variety of factors. Everything from institutional inertia to budgetary problems to manufacturing difficulties came together and combined with poor timing to really hinder efforts to develop a general-issue autoloading rifle.

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As far as institutional inertia goes, it helps to look back to the first generation of bolt-action rifles. Among the myriad of obsolete tactics that factored into the design of many of the early smokeless-powder rifles, perhaps the most strange to modern eyes is the magazine cutoff. The magazine cutoff disengages the magazine, making the gun a single-loader until directed otherwise by an officer (generally for emergencies). The rationale behind this feature was that armies up to that point had historically regularly had to work with strained logistics, and it was feared regular rapid fire from magazines would provide an unacceptable logistical strain on armies. Of course, WW1 quickly showed otherwise - railways were more than capable of providing supplies far more reliably than was expected. Nevertheless, the few wars between modern powers that happened between the invention of smokeless powder in 1886 and 1914 had provided some input away from more archaic tactics, and there was some degree of development towards semiautomatic rifles. France, for example, had recognized pretty quickly that autoloading arms were going to be the future, but they were hamstrung by their existing inventory - the 8mm Lebel cartridge was a poorly-designed stopgap hastily converted from an 11mm black powder cartridge, and thus an autoloading infantry rifle was going to require an entirely new cartridge if it was going to work well. For France, this wasn't a huge issue, as they already were ahead of the curve with the adoption of the Lebel, and felt comfortable enough to take their time perfecting an autoloader. Unfortunately for France, while this result in a modern cartridge and autoloading rifle to go with it (the Meunier), production only started in early 1914 and was dropped once war seemed imminent.

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I'm not familiar enough with other nations' attempts to develop pre-WW1 autoloading rifles to speak in detail elsewhere, but the trend is similar in that development was generally put on hold due to the high demand for rifles during the war. Wartime attempts generally were pitched as "conversions," like the British Howell automatic rifle and the RSC Mle 1917. The RSC 1917 actually holds the distinction of getting into service and seeing use on the front, but it was a heavily flawed design and was withdrawn from service at the end of the war. Despite that, most nations did recognize that semiauto rifles were the future, but there was also a host of other wartime developments that took development priority (light machineguns being one of the more significant ones). What that meant was most nations put development of a semiauto service rifle on hold while they spent their efforts elsewhere. France developed a new cartridge for a light machinegun in 1924, but had to redesign it into what is now 7.5 French, taking until the end of the 1920s to do so. Their long-term plans for rearming with a new semiauto recognized the logistical strain of rearming and compensated accordingly - a bolt-action rifle streamlined for production and use by second-line troops was developed first to get the older obsolete rifles out of the hands of everyone (MAS-36), while a semiauto was to come along later using most of the same tooling to equip frontline troops (MAS-40). Unfortunately for France, the MAS-36 wasn't able to fully replace older rifles before war broke out in 1939, and fewer than 100 MAS-40s had been produced before France capitulated.

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Elsewhere, we see similar problems. The Soviets spent quite a bit of effort to develop a semiautomatic rifle, ultimately resulting in several rifles nominally adopted in the late '30s (AVS-36 and SVT-38). Unfortunately, these had teething issues they never really got over, much of which was a consequence of doctrine and the cartridge being used - and although the Red Army had plans to replace the Mosin Nagant with the SVT-40, the outbreak of war in 1941 saw those plans dropped as demand for rifles skyrocketed. The SVT-40 itself suffered from a myriad of issues (including accuracy), and the Soviets would only finally adopt a semiautomatic rifle at the end of WW2 in the form of the SKS.

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The US stands out as the sole operator of a semiauto service rifle when it entered WW2 for a variety of reasons unique to the US. Rather than focus on light machinegun development after WW1, the US instead started trials for a semiauto service rifle in the early '20s. While these trials resulted in a lot of failed prototypes and some horrific designs like the Thompson autorifle, they'd continue on until the M1 Garand was finally adopted in 1936. Here, the US benefited from the small size of its peacetime army, meaning that it was fully reequipped with the rifle by the time it entered the war. Once war began, the Springfield Arsenal was helped by Winchester to keep up with production demands of the rapidly mobilizing army, meaning the US didn't fall into a situation like the Russians, where shortages forced them to revert to the old design.

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Other belligerents stuck with bolt actions out of either institutional inertia (Japan) or production issues (Italy and Germany). Japan notoriously took a lot of the wrong lessons from the war in China, and they entered WW2 lacking even a submachinegun. Japan did make some late-war efforts to develop a semi-auto service rifle, but they were barely able to keep up with their desperate supply situation with massively simplified stopgap bolt-action rifles, so rearming their soldiers was out of the question. Germany had several autoloading rifles in the war (G41, G43, and MP44), but the former two suffered from teething issues and were low on the priority for production, and the MP44 appeared far too late to become the standard. Italy also had some autoloading rifle projects, but they lacked the industrial capacity for more important wartime efforts, let alone a new service rifle, so these projects never materialized.