Were there any standardization of kit between axis forces or/and western allies like NATO has, were we can swap ammo, magazines, mg belts, gas mask, radio communication, jump cables and so on, or was it adapted later.
Briefly, no. British and American equipment was quite different. The rifles fired two different cartridges of two different calibers. The British .303 was rimmed and the US 30-06 cartridge was rimless dp they couldn’t be interchanged.
There were a few exceptions. Where different services used the same basic equipment (e.g. an M4 Sherman tank) one could exchange the large pieces but the fittings like the radios were often different.
In real life, British and American fighters of the same design could not be interchanged as they used different ejection seats, for example.
Most often the British and Americans fought in their own sectors and relied on their own supply and logistics organizations. They didn’t share supply trains as as more common today.
In three cases, Commonwealth and American troops fought in the same units and they generally standardized on American equipment as it was more plentiful in that location and they were the larger partner. The Americans and Canadians fought together in the Aleutian Islands and the Canadians used American.equipment. The US and Canada formed a joint battalion, the Special Service Force and US equipment was used.
MacArthur insisted Commonwealth troops use American equipment if Japan was to be invaded, but that operation was cancelled.
Were the commonwealth countries better standardized with each other?
The Commonwealth was more standardized, but not uniform, it also depended on service. Infantry weapons were standard, but uniforms showed a great deal of variety. Australia did not adopt the iconic battledress until after the war. Canada used a greener shade of cloth for battledress and kept the original, higher quality cut.
Air Forces and Navies were more standard as they involved high unit cost, low volume items like planes or ships, as compared to uniforms. Such items were either bought from Britain or directly licensed.
Australia manufactured its own fighter designs while Canada built foreign designs under licence. (That is a story I am sure Indy will get to).
The Commonwealth countries enjoyed considerable command independence. Canada had total legal independence, Australia was a halfway case ( another story for Indy, coming up shortly too) while the Indian Army was subordinate to British command but had its own equipment designs.
The Commonwealth was not a monolith but more like separate entities flying in close formation.