After WW2, the British developed the .280Brit (7x43mm Mk1Z) as a replacement cartridge for the .303Brit. The EM2 (No9Mk1) rifle was developed in conjunction with the .280Brit.
The UK was set on adopting the .280Brit and the EM2, but due to NATO standardization, they adopted the 7.62x51mm & L-1A1 (inch patterned semi-auto only FAL) instead.
During the Falklands conflict, both sides used variations of the 7.62x51mm FAL.
The first Gulf War, was the first time the UK military used 5.56x45mm rifles (SA-80 {L85A1}) in combat. After the Gulf War, due to constant reliability issues, H&K was consulted and rebuilt the SA-80 system. The UK military currently uses the SA-80A2 {L-85A2} as it's standard issue rifle.
During the 1950s, NATO decided to standardized to the 7.62x51mm as the main rifle cartridge. During the 1960s, most NATO countries had complied with the standardization and adopted 7.62x51mm rifles (mostly variants of the FN FAL or H&K G3).
During the 1970s, NATO decided to standardized to the 5.56x45mm as the main rifle cartridge. During the 1980s, most NATO countries had complied with the standardization and adopted 5.56x45mm rifles (almost all of which uses STANAG magazines).