During WW2 it seems that the Soviet military command quickly came to the conclusion that widely available weapons firing a cartridge lighter than rifle rounds, but heavier than pistol rounds and capable of automatic fire was crucial for modern warfare. Parts of the Nazi German military establishment seem to have reached a similar conclusion with the development of the 7.92×33mm Kurz cartridge and the StG 44 rifle.
Already in 1945, the Soviet Union introduced the SKS (their new standard infantry rifle) and the RPD machine gun, both firing their new intermediate 7.62×39mm cartridge. The SKS would soon be replaced by the famous AKM which was capable of automatic fire, unlike its predecessor.
On the other hand, the NATO states seem to have stuck much longer to the long, powerful rifle cartridges, the 7.62×51mm becoming the round around which new standard infantry rifles were developed.
The M14 - introduced in 1959 - was not officially replaced by the US military until 1970.
The L1A1 - introduced in 1954 - remained the main infantry weapon for UK soldiers until the 1980's
France changed to the intermediate cartridge-firing SIG SG 540 in the late 70's, only as a stop-gap solution before their new FAMAS rifle became widely accessible.
West Germany used the HK G3 - introduced in 1959 - up until the unification of Germany in 1990.
In short, the NATO states generally adopted intermediate cartridge-firing assault rifles decades after the Warszaw Pact countries. Why was this? Did NATO have a vastly different view of modern infantry combat than the Warszaw pact?
Most of NATO did see the value of a intermediate cartridge; right after the war Fabrique Nationale de Herstal (FN) designed prototypes of what would become the FN FAL that fired 7.92×33mm Kurz and .280 British. At the same time the British developed and even adopted as their service rifle the EM-2, a bulpup assault rifle firing the newly developed .280 British intermediate cartridge.
However these developments were put to a stop by the United States, being the largest member in NATO as well as supplying Western Europe with large amounts of economic aid they had a massive amount of influence over the armies of NATO. During the early 50s there was a push in NATO to have as much standardization as possible and everyone seemed to be moving over to the same standard in Europe, that being .280 British. However America would stand against this and due to their influence they won and forced the British to reverse the decision to adopt the .280 EM-2 and instead test a 7.62×51mm version. The 7.62×51mm EM-2 performed poorly in tests and was not capable of meaningful automatic fire, rifle cartridges were just too powerful for assault rifles. The EM-2 was dropped and faded to the footnotes of firearms historians. The FN FAL fared better in 7.62×51mm and was eventually adopted in some form by most of NATO (the British adopted it as the L1A1 lending itself to the name "Right Arm of the Free World". However this version could not fire automatically in a useful way and the L1A1 even dropped automatic all together.
There are multiple reasons why the Americans pushed for a full rifle cartridge over a intermediate one. First was the prevailing belief that sprung out of the American experiences in the Philippine–American War that a large powerful cartridge was needed as a man stopper; as such anything less would simply not do. The second was a prevailing culture of marksmanship that had persisted in the American Military since its inception. The Army put emphases on soldiers hitting their targets and devalued the individuals ability to fire automatically. Finally the Americans were so impressed by the performance of the M1 Garand in the Second World War they saw little need to adopt a new rifle for years to come, that's why they didn't adopt the M14 until 1959.
In short NATO didn't adopt assault rifles firing intermediate cartridges due to the Americans refusal to even consider intermediate cartridges even when many members saw the use and even the need for a select fire assault rifle that fired an intermediate cartridge.
Sources T.L. McNaugher "M-16 Controversy": "Bishop Chris" Guns in Combat : Thomas L. McNaugher "Marksmanship, McNamara, and the M16 Rifle"