There were several attempts by most major powers to adopt semi-automatic infantry rifles before or during WW2, but all but the US ran into problems that precluded them from replacing the older bolt-actions already in service.

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Since you brought it up, I'll start with the French. They had realized during WW1 that semi-autos were the future, but rather than continue the trend they'd been following with infantry weapons since the 1860s by continuing to adapt what was in service to new technologies, they decided to start over from scratch. The biggest liability from the start was their rifle cartridge, 8mm Lebel. It was a heavily tapered rimmed cartridge that was obsolescent in WW1 and made designing automatic weapons difficult. So they developed what became 7.5mm French in the early 1920s, which was an excellent cartridge well-suited for automatic actions. However, they focused development not on infantry rifles at first, but on the far more important Light Machineguns. They developed a modern light machinegun in 1924, but, as we saw all over the world, infantry rifle development and procurement would lull during this period. A semi-auto rifle was in the works, but it was low-priority until it was clear war was looming in the 1930s. What resulted from this design effort was a three-tiered system - a semi-auto rifle to be given to frontline troops, a bolt-action rifle for reservists that would share as much manufacturing machinery as possible with the primary rifle, and converted Lebels and Berthiers as a stopgap to allow the entire army to convert to the new service caliber quicker.

Unfortunately, this whole program kicked off too late to complete even the most basic stages - getting every soldier a rifle using the new 7.5mm caliber. The bolt-action rifle developed - the MAS-36 - would be arguably the best bolt-action service rifle ever developed, but infantry rifle quality generally isn't decisive in wars, and the majority of the French army was still equipped with a hodgepodge of Berthiers and Lebels either in the original 8mm or the new converted configuration. The semi-auto rifle, the MAS-40, was only barely rolling out pre-production models when the Armistice was signed. Once France was liberated, production was resume (with some major improvements to the design) as the MAS-44.

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The Soviets would be the closest to actually equipping the full army with semi-auto service rifles without ever reaching it. They had developed and adopted the AVS-36 in 1936, but it was a complicated design prone to a variety of issues. It would quickly be overtaken by the competing Tokarev design - the SVT-38 - in 1938, and the refined SVT-40 model would be adopted with the intention of making up a third of an infantry regiment's personal weapons by 1941. Unfortunately, this never came to fruition - the design was complicated to manufacture and difficult to maintain, and the outbreak of war saw factories either swapping back to manufacture Mosin Nagants or submachineguns, both of which could be manufactured far faster than the SVT-40. Additionally, the SVT-40 was far from satisfactory - attempts to press it into sniping, for example, generally fell flat, as the design suffered from inherent accuracy issues that made it significantly less accurate than the Mosin Nagant.

Once the Soviet position had improved and production was catching up with demand, it's likely the Red Army could have made another attempt to adopt the SVT-40, but the realities of war had changed. Volume of fire was found to be more important, so submachine guns were being deployed in much larger numbers. Meanwhile, the Red Army had recognized that the full-powered rifle cartridge the Mosin and SVT had been designed for was overpowered for the realities of combat. By 1943, they had developed a new intermediate cartridge, and semi-auto rifle development would shift focus to a new design - still using the Tokarev system - that would take shape as the SKS.

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Unfortunately, I'm not as familiar with Germany's circumstances, but their rearmament for WW2 was started from a much worse position than the other belligerents. Infantry rifles in general are much less decisive than other supporting weapons, so rearmament budgets were better spent on things like aircraft, armor, artillery, or machineguns. Germany's small arms development efforts seem to have been more focused on developing their general purpose machineguns than new rifles, and on paper it seemed to work out fine. The MG-34 and MG-42 would be fantastic weapons without a clear parallel among the Allies, so it's not unreasonable to say that gamble paid off in that area. They did eventually get around to looking at semi-auto rifles in 1940, but their first efforts would see similar thinking that other armies would fall into - things like requiring the barrel to be free from any holes cut in it (like a hole to divert gas for a piston) or a backup manual action. These would manifest in the Mauser and Walther Gewehr 41s, both of which would prove to be unsatisfactory. They'd be followed by the more reasonable Gewehr 43, which dropped the more absurd requirements and adopted the Tokarev system, but it was too little, too late. Even in 1940, it was too late to rearm with semi-auto rifles, and by 1943 there was no chance.

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Britain was a peculiar case. As far as I'm aware, they didn't have a major semi-auto rifle program during the interwar period like the other major belligerents did. In fact, their most significant semi-auto rifle program during the war would come out of Belgium - Dieudonné Saive and his crew from FN had arrived at the same solution Tokarev had concurrently developed over in Russia in 1937, and when their country fell in 1940, Saive and many other FN engineers would flee to England, where they continued their work. The British would help refine this design through the war, though it wouldn't come to fruition until after the war as the FN 49. However, due to the obsolete nature of the British .303 service cartridge, the rifle would be developed using the 8mm Mauser cartridge, even while under British control.

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In general, the chokepoint was production. Service rifles were generally last on the priorities when rearming for war, and it's a tall order to completely re-equip an army with new rifles. Perhaps the most telling thing is that even the US - the greatest industrial power of the war by far - had the M1 Garand adopted since 1936, but bolt-action rifles would still make their appearances with standard infantrymen as late as the Guadalcanal campaign in mid 1942.

More can definitely be said, but this older answer should be of interest.