Well the usual term is the "United Kingdom" which came out of the Acts of Union of 1707
United Kingdom Monarchs (1603 - present)
The Stuarts The Hanoverians George I (r. 1714-1727) George II (r. 1727-1760) George III (r. 1760-1820) George IV (r. 1820-1830) William IV (r. 1830-1837) Victoria (r. 1837-1901)
George III of the United Kingdom
"George III (George William Frederick; 4 June 1738[a] – 29 January 1820) was King of Great Britain and Ireland from 25 October 1760 until the union of the two countries on 1 January 1801, after which he was King of the United Kingdom of Great Britain and Ireland until his death. He was concurrently Duke and prince-elector of Brunswick-Lüneburg ("Hanover") in the Holy Roman Empire until his promotion to King of Hanover on 12 October 1814. He was the third British monarch of the House of Hanover, but unlike his two predecessors he was born in Britain, spoke English as his first language,[1] and never visited Hanover.
"His life and reign, which were longer than any other British monarch before him, were marked by a series of military conflicts involving his kingdoms, much of the rest of Europe, and places farther afield in Africa, the Americas and Asia. Early in his reign, Great Britain defeated France in the Seven Years' War, becoming the dominant European power in North America and India. However, many of Britain's American colonies were soon lost in the American Revolutionary War. Further wars against revolutionary and Napoleonic France from 1793 concluded in the defeat of Napoleon at the Battle of Waterloo in 1815."
Yeah, we all know the usual terms. My favorite is the United Kingdom of the British Empire. This evening was the first time I ever heard of the “British Union”. DL made it up in a desperate attempt to zeitgeist the Revolutionary War with The War of the Rebellion. He is attempting to unequivocally equate The Union, to the United Kingdom of the British Empire, in order to force his square argument into a round hole by his newly minted coined phrase. See my tag line.
If the need to "Preserve the Union" is the higher principle involved, then how could we have broken from the British Union?