EDIT My state was incorrect and the correct expression is ONLY nation-state as shown in the other comments on this page. ((Both are considered correct with "nation-state" being more British so as we are teaching AE we shall change the correct answer to "nation-state" with the other as an alternative. We accept all major variations of English.))
I beg to differ. I have never seen "Nation country" used in either American or British English. A quick Google search cannot find or define the term. Nation-state is correct English on both sides of the pond. This came up in review with tiles and expected Nation country. If nation-state has been marked as preferable, should this have happened?
I owe you and anyone else who has read my comment above a huge apology. You are absolutely correct and I can't think where I got that idea. (As an English teacher I should have known better.) I've added an edit to my incorrect post but haven't removed it so future readers understand the more recent comments. I've also removed any references to the incorrect phrase and left only nation-state as the accepted answer.
Many thanks for your correction and notification.
I have to search this term, because I didn't now that it existed such a concept. I let the link to the Merrian-Webster dictionary in case someone doesn't know that either: https://www.merriam-webster.com/dictionary/nation-state