I speak American English. My guess is that calling the UK a "country" would be seen as incorrect in British English. However, just about every map I see online showing "X by country" identifies the UK as one whole country, including those by the World Bank, Wikipedia, the CIA World Factbook...
Has the meaning shifted over time? What is the history of the word "country"?
The history of the word "country" is a research project, first published as A New English Dictionary on Historical Principles; Founded Mainly on the Materials Collected by The Philological Society, not just known as the Oxford English Dictionary. Originally, the word, from Middle French (meaning region, area of land delimited by natural or political boundaries) was used in a broader sense, as they say (sense 1) "The land of a person's birth, citizenship, residence, etc.; one's homeland", or (sense 2) "Land, terrain, or a region of undefined extent, esp. considered with regard to its physical characteristics" such as chalk country, fen country, stag-hunting country, country of the red deer. The latter use is fairly common in American English. I do not think the meaning has changed much over time, instead, politics has changed.
It looks like you can call the UK a country, judging by this sentence from Wikipedia:
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain,[note 1][19] is a sovereign country in Europe....
