The short answer to this question is complexity, semi auto rifles were significantly more difficult to manufacture, but also to maintain in the field, requiring much more significant logistical and training support given to soldiers, post Second World War every belligerent saw the benefit of semi auto rifles and began producing them and issuing them but that was with the benefit of hindsight, having seen how they performed in the war and in the context of the modern battlefield of the time.

The not as short answer changes per country, as you stated the US Army entered the war with the Garand as its standard issue rifle (although its worth noting that 2nd line units, reserve units and the USMC didn't receive the Garand until the war had already started, using the Springfield instead in the initial campaigns). The procurement program for the M1 Garand (or what would eventually become the M1 Garand) started in 1924, and it was adopted in the late 30's and the Army was predominantly equipped with Garands by 1942 (although again it's worth noting that deliveries of the Springfield continued until early 1944), so the US procurement program that allowed it to enter the war with a semi auto rifle took the better part of 20 years, and not every country had the privilege of expending resources on such programs.

During the same time period as above Germany was effectively crippled militarily by the Treaty of Versailles and when they began their rearmament programs the bulk of the focus was on tanks and aircraft, as you stated their infantry doctrine was built around the MG34/42 as the primary "punch" of the Grenadier squad (they issued their requirements for a semi auto rifle in 1939, Gun Jesus talks about the requirements in the first few minutes of this video)

The UK during this time was still dealing with the ramifications of the First World War and while it did develop advances in weapons there was little appetite for completely changing the standard issue weapon of one of the most spread out and varied standing armies in the world (in particular the range of environments was a concern), similarly to the Germans the British (and Commonwealth) built their rifle sections around the Bren LMG, who's role it was to suppress the enemy while the riflemen were there to fire accurate shots at long distance (automatic weapons were present in the form of submachine guns but British doctrine heavily emphasised accurate rifle fire at a distance as the best way to deal with an enemy force), for their doctrine the SMLE was perfectly suited to what the British wanted from their standard issue weapon (and this remains' the case even today, where the British Army emphasises marksmanship and accuracy over other aspects of weapons)

The Russians began issuing self loading rifles during the First World War but national upheaval during the Revolution stalled their programs, they ended up on a similar track to the US procurement program (with troops armed with semi auto rifles fought during the Finnish War in 1939), the intention was for the entire Red Army to be equipped with the SVT38/40 however the German invasion in 1941 delayed those plans. Again repeating complexity, the SVT was a good rifle but by the end of the war there were 12,000,000+ troops in the Red Army who required weapons and the Mosin was far simpler to produce than the SVT and therefor remained the standard (although several million SVT's were manufactured and were relatively common throughout the Red Army) (Again you can watch more about the procurement and the rifles in detail from Gun Jesus)

As you say in your OP Italy, Japan didn't really have the industrial base to mass produce semi auto weapons (or to conduct the research required to build them), I don't really know much about France so I can't comment on their efforts.

In summary the complexity of semi auto rifles in both manufacture and maintenance combined with an "if it isn't broken don't fix it" attitude from many militaries (particularly on the Allied side as they won the last big war with these weapons) lead to several major nations dragging their heels when it came to semi auto rifle procurement, however once the advantage became clear during and immediately after the war it was a priority for most nations to acquire semi auto rifles (the UK had the L1A1 (FAL) in 1954, the Russians introduced the SKS in 1945 and then converted entirely to the AK47 in the late 1950's (Again this is a good example of the complexity of the semi auto/assault rifles as even copying the STG-44 it took the Soviet Union the better part of 10 years to introduce their own version)

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TL;DR It's hard to design, hard to manufacture and hard to maintain semi automatic rifles at the scale that the 2nd World War was fought at, and the only country with sufficient industrial capacity to do it was the US, but the Red Army and the Wehrmacht had significant numbers of troops armed with self loading rifles by 1945

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Edit: I've included a couple of Youtube videos if you want to hear more about those specific weapons mentioned (I do recommend them they are full of useful tidbits) but here are some actual sources

General production and design of self loading rifles - Jenzen-Jones, N.R. (2017). "Global Development and Production of Self-loading Service Rifles, 1896 to the Present"

Production/Design and use of the Springfield/Garand - Bruce Canfield's U.S. Infantry Weapons of WWII

Russian Small Arms references - David Porter Russian Weapons of World War II

Bolotin, David Naumovich (1995). Walter, John; Pohjolainen], Heikki (eds.). Soviet Small-arms and Ammunition

German Small Arms - Self loading rifles in particular

Overview of German small arms as a whole

General Overview of Allied Small Arms

Here is an interesting infographic website with information on a range of different small arms from the 1930's (They don't cite sources but it's a cool infographic/website nonetheless)

More can always be said on the matter if anyone would like to address the matter of semi-automatic weaponry in the interwar period; for the meantime, as this has come up (milhist is like that), here are some previous posts for your perusal:

• u/Georgy_K_Zhukov has an overview of the major combatant nations and their difficulties;
• and u/Meesus looks at France specifically.

Agree with the sentiments that have already been posted throughout the post, and wanted to add a couple additional notes.

• If you have a weapon that can fire X rounds a minute, that means that you have to have the manufacturing capacity to produce that many rounds and the transport capacity to get that many rounds to the front. Infantry rifle rounds would be competing for space on trains/trucks with fuel, ammo for tanks, artillery, and machine guns, food, medicine, reinforcement soldiers, etc. I'll give one example of how this can be an issue. Great Britain, which had a military machine that could support an empire all over the world, had a full-blown Shell Crisis in 1915. They had been experiencing shortages of shells since the autumn of 1914, and they had only started combat operations at the end of August ... with 100k men, in a front that, though separated from England by the Channel was still physically close. If you increase the firing capacity of a single rifle, and give that rifle to hundreds of thousands or millions of men, and push the front lines far from your industrial core (say from Germany to the Russian hinterland) the supply issues increase exponentially.
• Many militaries had adopted doctrines of combined fire between the wars. During WWI, they'd learned that single weapon systems couldn't handle all the jobs being handed to it. The French overproduced the 75mm field gun, and it didn't have the ability to fire in a way that would be effective against entrenched enemies - they needed a howitzer that could fire in a high arc (like lobbing a hand grenade) and the 75mm could only fire at a narrow range of inclination (every pitch was a fastball). Same thing with bolt action rifles - they were great at one job, but couldn't perform others. Over the years, they kept the bolt action rifle because it worked well in a narrow role, and they added additional weapons that were better regarding short-range situations, rate of fire, or special use/need. Submachine guns, antitank weapons, flamethrowers, light machines guns, grenades, and sniper rifles were added so infantry were more of a Swiss Army knife. Infantry might not be the best tool against everything (example: tanks) but with the right weapons in hand they could slow something like that down until they got their own special weapons into position to counter it.
• In a conscripted army, it sometimes helps to have a limiting factor on a weapon to account for inexperienced soldiers. If you take a raw recruit and hand them a machine gun, they'll run through a week's worth of ammunition in the first encounter with the enemy (or the first time a twig snaps), likely hitting nothing. A rifle that fires one round at a time and has an easy but set manual reloading procedure between each shot can help ease a raw recruit through the first initial shock of combat without them wasting ammunition. In stressful situations, you always go back to your training, and the bolt action keeps you connected to that training because there aren't any shortcuts - you have to reload it exactly as you were trained between each shot, aim like you were trained, etc. Bolt action rifles were simple but rock-solid and only had a few parts that could break or jam. If your brain is in a state of shock and not working at 100%, you need a weapon that will work at close to 100% to cover for you.
• They were a good investment for hampering operations and lowering morale in occupied territories. A bolt action rifle in the right hands can hit anything the human operating it can see, so a rifleman or partisan or resistance fighter can hide almost anywhere for a shot. And they don't have to stay there until the job is done - you can stay there and fire one or two shots before melting away. At the cost of a couple bullets, the enemy needs to use up time investigating for snipers, and enemy morale is lowered because soldiers are on edge for being in "sniper territory." Interrupting the enemy's timetable can have some real knock-on effects, and you can manage that with a bare minimum investment.
• Added bonus - these weapons were so simple that, with a crash training period you could throw these weapons at inexperienced people and they could operate them with at least a below-average level of competency. This was handy for the Soviets who sometimes had to pull citizens out of their homes or workers out of their factories to slow down the blitzkrieg or for partisans in occupied territories. Any civilian in your territory is a potential soldier in an emergency situation, any civilian in occupied territory is a potential partisan.

They were antiquated, but there was still a role they could play. Something old and useless you can toss out. Something old but still useful you keep around.