Timing and circumstance more than anything. Had war broken out a few years later, the USSR and the French probably would have been soldiering with semi-automatic rifles as their standard issue arm.

The United States adopted the M1 Garand in 1936, and it entered production a year later in 1937. By the time America entered the war, conversion from the M1903 Springfield had been completed by the US Army (The Marines were slower to replace the Springfield and still used them in the beginning of the war). I start there to point out that the United States was a) an industrial behemoth, b) Remained out of the war until late 1941 and c) Employed a small, professional military during the 1930s, yet it still took them four years to change over. They weren't the only country looking into the possibility, just the only one who was able to do it.

France adopted the MAS 36 in 1936 as something of a compromise. Semi-automatic designs had been worked on through the 1930s and earlier even (See the St. Etienne M1917 for instance), and originally they were hoping to adopt a semi-auto by 1940. The MAS-40 would have been adopted had the war not interrupted things, and the MAS-36 would have been seen as nothing more than a short time placeholder. Its design even reflected this, as my understanding is that the receiver imitated the semi-auto designs closely enough that the same machinery could be easily switched from making MAS-36s to MAS-40s.

Likewise the Soviets had been perfecting their own semi-auto designs, first with the problem plagued AVS-36 (which actually was full-auto capable, but lets not quibble!), and then the SVT-38 and SVT-40, the last of which was considered a very solid design, and had gone into production well before Operation Barbarossa. In fact, it was slated to replace the M91/30 Mosin Rifle around that time with production of the latter being scaled down, but again, the exigencies of war meant that the USSR simply couldn't afford to continue with that plan, and instead ramped up production of the Mosin, although the SVT was made in large numbers along side it.

So what this all is to say is that America wasn't alone in its quest to create a semi-automatic service rifle through the 1920s and 1930s. The shortcomings of the bolt-action was becoming apparent, and other countries looked into the possibility. But the design process was slow (the M1 project dates to the 1910s!) and production doesn't start off with a bang (In the US they were building 20 M1s a day in March of '38, and about 200 per day by Jan. 1940. Official estimate was 145,832 needed for total replacement, obviously much smaller than the European armies. That number of course rose once the peacetime draft began in late 1940). If World War II hadn't happened, or at least their production capabilities considerably greater than in reality maybe (?), at least France and the USSR would have been fielding semi-automatic rifles as their standard service arm by the mid-1940s.

Germany did have limited testing in the '30s of an automatic carbine using an intermediate cartridge, the Vollmer M35, which was actually considered an excellent design, but few recognized how useful it could be, so development died and it wasn't until after the war began, that a domestic semi-auto rifle program began, resulting in the G41/G43. It is also worth nothing that the G41 was considered something of a dud due to rather heavy requirements on the design (There were actually two versions, Mauser's being REALLY bad and Walther's being just mediocre). The G43 was a considerable improvement, building off the Walther design with some influence from the much better Soviet SVTs they had encountered.

Even if development had started earlier though, keep in mind that the German Military was expanding rapidly in the late 1930s, from a small force capped at 100,000 men into one of many millions. Starting up production of an entirely new weapon would have been a rather significant undertaking and hampered their ability to quickly rearm.

As for other major powers, such as the British or Japanese, I don't know of any serious program that otherwise would have come to fruition prior to the war breaking out. The Japanese late in the war made a small number of Type 4 Rifles, which were little more than M1 Garand clones, but that didn't start until well after war began, and I don't believe were ever issued.

So anyways, that is the sum of it. If you want a TL;DR, I guess it would be "Retooling production for a totally new weapon to rearm millions of men with in the middle of a war is kind of hard to pull off".