'Zjednoczone Królestwo' is correct, it's a literal translation, but absolutely NO ONE would use that in speech. It's like using fullnames of countries all the times. You don't say 'The United Kingdom of Great Britain and Northern Ireland' every time. It's always "Wielka Brytania" in Polish. My brother lives in UK = Mój brat mieszka w Wielkiej Brytanii.
United Kingdom in Polish is Zjednoczone Królestwo.