I get the feeling that NATO tends to drive a wedge between between First and some ex-Second World countries.
What do you mean? The "Second World" is the Americas.
In an outmoded Cold War context, some Americans use the term First World in reference to the arguably "capitalist" countries advocating liberal or social democracy, Second World as a label for countries deemed "communist" advocating Marxist-Leninist ideologies, and Third World to describe unaligned, mostly developing countries not an integral part of the geopolitical struggle. Many folks might have a different understanding of it though. In the States, I reckon the Old World refers to Europe, Asia, and Africa while the New World is in reference to the Americas.