The British Empire was defeated because it asked the Americans to turn against their own neighbors. The British, French, Spanish, and Dutch were all colonizing the Americas at the same time and many had lived side-by-side as neighbors for hundreds of years. When the Crowns of Europe began to pursue 'nationalistic' policies calling on their country men to kill off those from other countries they had come to know as neighbors, those people, led by the British colonialists who were guided by the principles of the Union, joined together to cast off their former Kings.
And yes, GB was cast out of the Union, but are still with US in spirit see attachment: