The two most prolific rifle cartridges in use by NATO are the 5.56x45mm (derived from a .22" cartridge) and the 7.76x51mm (derived from a .30" cartridge). The Russian 7.76x39mm is also derived from a .30" cartridge. However with the exception of the USA, all NATO nations use a metric system, and have for decades. Russia adopted the metric system in 1925.
So, why do all these nations use ammunition with "odd" measurements (in metric) derived from imperial measurements? Why is the 5.56x45mm not 5.6x45mm? Why is the .30 Russian not a 7.8x39mm round? Do the extra 40 microns really make that much of a difference in the performance of the firearm?
The Europeans quite early used metric for specifying their cartridges, the British and US used Imperial, and then after NATO was established and had decided on standard ammunition, that standard ammo was defined in metric. So, .308 Winchester of 1952 became 7.62 NATO in 1954, the .223 Remington of 1957 became the 5.56x45 in 1964. Though the process of NATO deciding what cartridge to use was quite political, that imperial-to-metric bit was simple. However, there are actually three ways to specify the caliber of a gun: there's the diameter of the bullet, the diameter of the bore defined as land-to-land of the rifling, and the groove-to-groove diameter of the rifling. This is how you can have .38 Special and .357 Magnum being the same diameters- .38 groove to groove, .357 bullets.
This introduces a wackiness to the system right off. But perhaps it would be easier if you simply started with the premise that these are all just names, whether metric or imperial, because even the numbers are not precise. So, 7.62X 54 Moisin Nagant has bullets that are .310" in diameter, or 7.874 mm. land-to-land bore diameter of 7.62mm. The SKS 7.62x39 has bullets .311" , and so 7.8994mm, but still a 7.62mm bore. The US .30-06 is called 7.62x63, has a 7.62 diameter bore- but it's got.308 diameter bullets. It seems it ought to be, then, the bore diameter that we think of. But that's not always quite true. The Japanese 7.7mm Arisaka had bore diameter .303, which is really about 7.7mm , yet that's seems to have the same bore as the 7.65mm Argentine Mauser. And the K31 Swiss Schmidt-Rubin has bore varying from 7.52 to 7.54mm, even though the cartridge is called 7.5 x55.
As to why so many countries would have high-power rifle cartridges around 7.62 mm is another question, involving tradeoffs of flat trajectory vs barrel life, range vs ammunition weight. 7.62 was a pretty good compromise when it was thought longer-range accurate shooting was very important: when it was decided that lighter high-velocity ammo at shorter range was more important, 5.56 bullets were found better.
Barnes, Frank C ( 2006) Cartridges of the World