Do NATO Collective Defense obligations cascade when a member goes to war due to other defensive treaties?

What would happen if a NATO member was compelled to go to war because of a separate defense pact they were a part of?

As an example, Russia has threatened Sweden and Finland if they try to join NATO. While these 2 countries aren't part of NATO, they are part of the EU's mutual defense clause, and the EU contains NATO members. I would assume that, should Sweden be invaded, France, Germany, and other European NATO members would be obligated to respond. Would this, then, involve other NATO countries outside the EU, like the US and UK? Obviously there would be plenty of political pressure to get involved, but I'm interested in the question of what the NATO treaties require and/or encourage.