No other nation equipped their armies with semi-automatic rifles to the extent of the Americans - this was due to a multitude of reasons.

There's a fundamental difference in European and American infantry rifle squad tactics during ww2.

The Americans believed the rifle squad could provide its own covering fire with semi-automatic rifles and neglected machineguns, light, medium and heavy. A German 1943 Grenadier battalion had 44 MGs (MG 34 and/or MG 42) while a US infantry battalion had 8 (35 if you count the BARs, but without interchangable barrels, they were not capable of sustained fire). All the US machine guns were mounted on tripods, making them heavy and hard to move with the infantry. Generally, I feel the sentiment both by historians and the US army post-war was that this doctrine was wrong - a rifle squad was generally unable to provide its own coverering or supressing fire, and the US army introduced MGs on the squad level.

The US did introduce the M1919A6 as a belt-fed bipod mounted squad MG for its paratroopers, but not for the general infantry. Immediately after the war, the they started to develop a squad MG and introduced it in the belt-fed M60 in 1957.

As you can see, the Germans focused on the belt-fed MG and the sustained fire it could provide. The British focused on heavy watercooled HMGs (Vickers) and light magazine-fed LMGs (Bren). They used their heavy MGs with advanced tactics developed during the latter part of ww1, setting them up to fire indirectly and cover areas with fire to deny them to the enemy (they could not move in the bullet-riddled area) or keep enemy soldiers down while their own soldiers advanced.

That said, while the Germans were impressed with the quality and performance of the M1 Garand, it did not fit into their own unit tactics - they did supply sharpshooters with their own semi-automatic Gewehre 43 rifles but did not aim to supply the general infantryman with a semi-automatic rifle. Instead they, from experience saying the combat distance of infantry was almost always below 300 meters, they attempted to merge the SMG and the rifle into the assault rifle - their MP 43/StG 44.

Once the Germans encountered large amounts of US troops equipped with semi-automatic rifles, they had already focused on the heavy general purpose MG (MG 34/MG 42) and SMGs and later merged the rifle and the SMG into the assault rifle.

The full power semi-automatic rifle as a general armament for all riflemen remained a US curiosity, even after the war with the US M14.

The British focused on Bren LMGs (up to two per squad) and SMGs ot increase the firepower of their squad. After the war, they eventually replaced the rifles and the SMGs with the L1A1 assault rifle in 1957.

The Germans focused in general purpose MGs (MG 34 and MG 42) and SMGs and then merged SMGs and rifles into the MP 43/StG 44.

The Soviets focused on SMGs, creating an entire company in each rifle regiment equipped with only SMGs and LMGs. They also used a lot of LMGs (DP-28). After the war, they went the same way the Germans had, merging the SMG and the rifle into the assault rifle, in their case the AK-47.

As you can see, while other powers might have been impressed by the M1 Garand, it did not fit into their infantry tactics and doctrine, and viewed over the world and historically, it was a dead end only used in the US.

Nomenclature:

SMG = Sub-machine gun.

MG = Machine gun.

LMG = Light machine gun (most often magazine fed).

GPMG = General purpose machine gun (on bipod, belt-fed).

HMG = Heavy machine gun (on tripod, most often belt-fed).

The only Axis nation I know of that felt the effects of the Garand and decided to implement a version like it were the Japanese. They created a ten round rifle called the "Type 4". It was essentially a carbon copy but with a magazine inserted underneath, rather than a "clip" inserted at the top.

The Germans had not experienced the M1 in combat until 1941/2, but after France Hitler understood the practical benefits a semi-automatic rifle could have in infantry units. He ordered the production of the "Gewehr 41". This was a complete failure, regularly breaking due to an overly complex design. This was manufactured with a bolt action and semi-automatic system in place, so it could be used as both if one failed. This was also capable of being loaded with a magazine underneath or with two Karabiner 98K "stripper clips" directly into the top. Being a complete failure, it was later redesigned as the "Gewehr 43", a much better weapon.

So, although they originally designed a semi-automatic weapon before American's fought them, the German's definitely realized the implications of having a rapid fire rifle in combat.