You're right about the special relationship between the two countries as well as the U.S. and the U.K's commitment to NATO, but there is a misconception here in regards to NATO.

Article 6 of the treaty specifically states that member nations are only obligated to come to the aid of a member if that attacked territory is north of the Tropic of Cancer. The Falklands are obviously well south of the Tropic of Cancer, so that explains why NATO's involvement was so curtailed. Though it should be mentioned, the U.K. did receive various forms of assistance from multiple NATO members, though that assistance was not directed through NATO itself.

As far as the U.S. not being involved or helping the U.K. in the Falklands, they did, but only on a limited basis. The U.S. unsuccessfully attempted to mediate the conflict between Argentina and the U.K.

When that failed, Secretary of State Haig made it clear the U.S. would prohibit arms sales to Argentina and publicly announced the U.S. would provide the U.K. with material support. One of the items initially promised to the British was the use of USS Iwo Jima (at the request of the Royal Navy) in the event that they lost either of their carriers (HMS Hermes or HMS Invincible), which was to be crewed with military contractors (though how many and where they would be found in short notice was never fully explained). The U.S. also provided other material support, chiefly with equipment and the use of U.S. operated airfields at Ascension Island so that the British could carry out long-range sorties/reconnaissance flights over the Falklands.

Norway and France both provided the UK with vital intelligence. Norway devoted one of their SIGINT (signals intelligence) stations to help acquire satellite imagery of the South Atlantic at a time when neither the U.S. or the U.K. had any satellites over the area. France had HUMINT (Human intelligence) teams operating in Argentina to gather intelligence as well as to help forestall the delivery of Exocet anti-ship missiles, though there have been unsubstantiated claims the French had technicians in Argentina assisting the Argentinian's with the use of Exocet missiles.

Indeed, article 5 of the North Atlantic Treaty does state that an attack on one is an attack on all, however article 6 specifies that it's only good for attacks north of the Tropic of Cancer and what was at the time French Algeria.

The US were at the time propping up some pretty unpleasant regimes (either far-right or outright military dictatorships) across South America (Operation Condor), most notably (in this case) Argentina, as part of the cold war with the USSR (it was notable that the main people that these regimes were repressing were left-wing).

The US found itself allied to both parties in the Falklands and tried to act as a "neutral" go-between to negotiate a peace before the British Task Force arrived, much to the annoyance of the British. However, this position gradually changed as the fleet approached the Falklands and the position of the Argentine Junta repeatedly changed between the departure of the US envoy from Buenos Aires and his arrival in London to support the UK.