NATO does exist in Fallout, however the person you replied to was a bit erroneous in the timeline. Canada allowed US troops on her soil to safeguard oil pipelines, however those troops were used heavy handedly to do more than that, and increasing protests/riots regarding US troops as well as an sabotage attempt on a pipeline became the justification for annexation. The lore afaik doesn't go much into how NATO or other militaries were involved in the Sino American War.

Presumably NATO was effectively just the US and Canada at this point, Europe had been decimated with its war with the middle east which included a nuclear exchange, and the United Nations disbanded. Under these bleak circumstances, one can see how Canadians might chafe under US military occupation.

Also note that Article 5 does not expressly mandate the level of military support an individual member must provide if it is invoked, indeed it is up to the individual state to take such action as it deems necessary.

Whether or not Canadians contributed much support on the battlefield in Alaska or mainland China is somewhat immaterial, it was the United States whose recently developed power armor that turned the tide of the war. Canada's main role was possession of natural resources useful for the war effort.