The problem with a global nato has always been two fold in my opinion. Firstly the unwillingness of most of the nations in places like Asia, Africa, and Oceania to commit to a nato style alliance and secondly the unwillingness of smaller nato countries in Europe to defend them. America was being a world power definitely had an interest at forming more secure defensive pacts with their allies in different parts of the world examples can be seen in the now defunct South East Asian Treaty Organization and the proposed North Pacific Treaty organization. These were created/preposed mainly to create a coherent mutual defense structure among American Allies In the Cold War as well as to contain communism. I think the most realistic way to make a global nato is it for to act more like a few little defensive pacts for each area of the world like SEATO was for south east Asia and NATO is for Europe. Most European would not want to defend nations on the other side of the world if they got invaded and Vice versa which makes this system where America (and possibly Canada) which as mentioned has interest in defending all its allies around the world the common pillar in each alliance.
Now with that rant done how to acually accomplish this? A more successful SEATO seems like the best way to me let’s say America is able to keep the pact coherent and together it could inspire more similar pacts to form as the first 2 we’re so successful at securing American interests. I think we could see a North Pacific pact with America, Japan, South Korea and some of the smaller island nations form it basically already exists in theory and so it would be easy to make formally real. The other one I think might form would be a Middle East pact in reaction to the Islamic revolution in Iran. There was already some attempts to make a NATO style defense alliance by the British after the Second World War and I think that a nato style alience including Saudi Arabia, The gulf States+Yemen, Jordan, and Israel