Sometimes you find answers in unexpected places. I found the most succinct description I've ever seen of how England, Great Britain and the United Kingdom relate to each other in a library newsletter about travel guidebooks:
London is the capital of England.
England is a country.
Britain is an area that consists of England and the country of Wales.
Great Britain is the name of the island that is home to the countries of England, Wales, and Scotland.
The United Kingdom (UK) is a country that is a union of the countries on the island of Great Britain, along with the country of Northern Ireland (which shares the island of Ireland with the Republic of Ireland.)
[The Republic of Ireland is] a separate country that is not part of the UK.