Nevertheless is not something I'd expect in a casual conversation, but it's heard in business meetings, negotations, sermons, presentations. I wouldn't call it archaic at all.
Notwithstanding, in my experience, is heard more rarely and has a "legal" sound to it. I'd expect to see it in contracts, lawsuits, and government publications. I would be very surprised to hear it in a conversation, although I can imagine a politician using it in a speech.