British forces had actually started to develop and, in 1951, to introduce an automatic rifle, the EM-2. In fact, this was a rather innovative weapon for its time, featuring both a bullpup configuration and an intermediate cartridge. However and despite the good experience of the Germans in WWII with their intermediate round, the US Army back then deemed intermediates to be too weak and insisted on standardisation of the existing and far more powerful 7.62×51mm round for NATO. In return, the Americans promised to standardise on the rifle design as well, for which they agreed to the Belgian FN FAL.

So, the UK had to scrap its EM-2 for the Belgian FN FAL, which was modified for imperial units and introduced as the L1A1. The Americans in turn went ahead with their eventual M14.

Ironically, the L1A1 was itself replaced by a bullpup rifle with an intermediate cartridge, the SA80.

Source:

David Westwood: Rifles: An Illustrated History of Their Impact. Santa Barbara: ABC-CLIO, 2005 - ISBN 1851094016

Actually the Mexicans lead the way by adopting the Mondragon back in 1908 were it served alongside the older straight pull bolt action rifle. It first saw combat during the Mexican civil war and during ww1 these rifles found their way into German hands (As they were produced by SIG and the Germans bought them) were the infantry tested them but found them to be to sensitive to dirt and mud so they rejected them. The German Air Force on the other hand didn't have a problem with mud and dirt so they adopted them for use in air combat and they found that the rifles were a great improvement over the earlier Gewehr 98. When enough synchonising gears were made to equip all fighters with synchronised machineguns the rifle fell out of use and most seems to have been destroyed or given to the navy.

The Germans developed a self loading rifle during ww1 but again because it was unreliable in mud and dirt the infantry rejected it and the Airforce adopted it during the brief time when air combat was about shooting the other plane with small arms. After the war the Germans had a case of the Versailles treaty so they didn't do much more with self loading rifles until 1940.

The French were also very early adopters of semi auto rifles and their first production model was the Fusil Automatique Modèle 1917 which entered service in 1917. The soldiers didn't really like it though since it was long, heavy, and had reliability issues in the trench warfare of western europe. It was improved in 1918 and was planned to replace all bolt action rifles starting in 1919. I'm not sure why they didn't go ahead with the plan to replace all rifles (they didn't even replace all existing self loading rifles) but a semi educated guess would be that peace got in the way of progress as always happens.

The US were interested early on before ww1 but nothing really came of their designs until after ww1 when they started experimenting with different self loading rifles. And it wasn't until the early 30s when they started getting serious about developing a self loading service rifle. And the Garand entered service in 1937 but it wasn't until 1944 when the Garand had been issued to all front line units and the Springfield wasn't completely replaced by the Garand until some time after ww2.

The Russians did some early work before ww1 but because of political instability they didn't produce their first self loading rifle until the early 30s. In 1936 they adopted their AVS-36 which was actually not a self loading rifle but a select fire battle rifle. Then they went on to produce some pretty good self loading rifles during ww2 (Their experience with the AVS-36 showed them that battle rifles are a silly concept).

Now the part you are probably most interested in The British. The first British self loading rifle was the Farquhar-Hill which was patented in 1908 it went through a major revision in 1915 and it wasn't until 1918 that the British army ordered the rifle but when the war ended the order was cancelled without any rifles being delivered. After ww1 the plan was to replace the .303 with an intermediate cartridge and to adopt a more modern self loading rifle but due to budget constraints the armed forces in general and the army in particular were severly hampered in their ability to adapt with the times and become ready for the next great war. And by the second half of the 30s when a new war seemed imminent they stopped a lot of development and focused more on rearmament with the stuff they had at hand.

After ww2 the Brits, just like a lot of other nations, stopped focusing on developing self loading rifles and started looking into the Sturmgewehr designs that the Germans had and the Brits went back to their pre ww2 ideas of an intermediary cartridge except this time in a select fire assult rifle. By 1951 their design was finished and they started producing the EM-2 bullpup assault rifle chambered in the .280 intermediary cartridge. But in the same year NATO decided that their standard cartridge should be the full powered 7.62x51 and so the EM-2 was cancelled and the British army decided to simply buy a battle rifle of the shelf.

TL;DR: They adopted one during ww1 but peace broke out and it was cancelled. They spent the interwar year being chronically underfunded which hampered their development projects. During ww2 they focused most of their energy on stuff they had available and could produce right now. After ww2 they simply skipped semi auto/self loading rifles and went straight on to the assault rifle. Then NATO intervened and forced them to adopt outdated technology.

Edit: Also note that when the British army adopted the FN FAL (L1A1) they opted for a semi auto version only because they realised that the full power 7.62mm cartridge is way to powerful for your average soldier to be able to control during full auto.

This old answer about WW2 addresses why, in terms of squad tactics, the bolt-action rifle was in some cases preferred to the semi-automatic rifle, and the way different armies also equipped squads with with submachineguns, machine guns, and artillery. I think you will find the discussion interesting, though it is a little before the period you are specifically asking about. :)