US, and always was diapers. An odd thing to me, nappies (in the UK, and down under) were usually terry cloth. Here in the US though, terry cloth was not a common diaper material, usually Birdseye cotton. However, here in the US training pants, the ones with plastic outer, were terry cloth inner.
Being born and living in the USA, I've always referred to them as diapers. I too once heard many years ago that "Nappie" was evolved from "Napkin". I use a napkin to wipe my mouth when eating, often the same one several times during my meal. I wouldn't want to wipe my mouth with a diaper, especially a used one!