The 5.56 mm Cartridge was officially adopted by the U.S. Army in 1964, since the fallout time line splits right after WW2 why do we see so much 5.56.
this seems like a pretty big mistake if you ask me

Fallout 3 has a lot of errors when it comes to militaria/firearms....and your right the ammunition in question should be 7.62x.51 NATO

Why should the 5.56 mm standard not have been introduced in the Fallout universe ? Its development was not directly related to international circumstances. Only its standardisation was.

Interesting fact : The 5.56 mm round introduced by the USA in 1964 is NOT the 5.56 mm round that was standardized for all other NATO countries later on.
The 5.56mm NATO standard is, or better said was modeled after the SS109 round from belgian manufacturer FN which utilized a heavier projectile with steel core.

Today, Germany for instance doesnt use the SS109 standard anymore but instead a dual-core ball 5.56mm cartridge which combines armour piecering and in-body fragmentation properties in one type of ammunition. It is also the first 5.56mm round to utilize a new kind of non-toxic propellant.

The Assault Rifle is based on the G3 as well.

Furthermore, the 5.56 is listed as 556 anyway.

Bullets have got to be some size... Theres no reason why parelell development couldnt happen.