Also a BIG assumption, I would say a false one, to assume that NATO would all band-together to fight China. If that logic were true, World War III would have started when Russia invaded Georgia - and it did not. Why? Because there was no power capable of stopping Russia with a country on its boarders.
You obviously misunderstood what NATO actually is. It is a defence alliance formed in 1949 to deter the expansionist Soviet Union and if the deterence should not had worked, to have posed a big enough military power to effectively fight the Soviet Union's military forces in Europe.
It is a defence alliance. The contracts say that a non accidental military attack or a war declaration on any of the member countries is considered a attack on all member countries. Its as simple as that. Thats the main point of the deterence.
Thats a binding contract. There is no "Oh we changed our mind" option in those contracts otherwise it would be worthless.
Meaning you declare war on for instance Poland , you declare war on 27 other NATO nations at the very same moment and they declare war on you within 24 hours. War between member states is fully out of the question and forbidden. There is even a paragraph in the contracts that clearly says that.
Georgia ? Thats a totally illogical example as Georgia is not a NATO member therefore there was no obligation to come to Georgia's help for anyone.
Its a partner of NATO's Partnership for Peace program which is a sole diplomatic construct for coordinating diplomatic approaches and nothing more beyond that. A thing Georgia ignored when it tried to bring South Ossetia back under its authority by using military force.
In case of war this PFP means absolutely nothing except that diplomats of NATO countries will be involved in negotiations like a cease fire. There is no half-membership or anything like that.
Things would had been different if for instance NATO soldiers would had been killed in russian attacks on Georgian soil.
Of course here things would be difficult as you would need definite proof that this attack was intentional and not a accident. As good as impossible except Russia would say "Yes we knew it were american soldiers and it was our intention to kill them". That would be the point when world war three would start.
So in a Fallout universe with NATO in existence, the invasion of Alaska would had triggered a war declaration of entire NATO on China. This is THE role example of the case of common defence in the NATO contracts. There is no room for flexibility under these clear as can be acircumstances that a member is under attack by another country.
Nonetheless there is not the slightest reference to that this happened nor do we have any hint of european soldiers fighting in Alaska.
For as long as the opposite isnt stated in a Fallout game these are the best indications that NATO did not exist anymore at that time , maybe never existed at all or the USA was not a member anymore
The only other defence alliance of this kind is the WEU "West European Union" which is a seperate non-NATO related common defence contract between the UK, West-Germany ,France ,Italy,Belgium,Luxembourg and the Netherlands signed in 1954. The contract automatically adapted to the reunification of Germany and is still in effect today although its relevance has faded for obvious reasons.
Its pretty much identical to the NATO contracts involving a common defence by explicitely mentioned "all available military means" in case one of the members is being attacked. The background to this contract was also the threat posed by the Soviet Union.