Well, this case is very similar to what happens to the USA. A lot of times, you would hear people refer to the USA as "America," right? It's the same issue with England and UK.
So with that in mind:
United Kingdom (UK) = United Kingdom of Great Britain and Northern Ireland
Great Britain (Britain) = England, Scotland and Wales (part of United Kingdom)
England = a country (part of Great Britain, which is part of UK)
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