Calling the whole show England is not only incorrect, it irritates people.
I didn't make this usage up. Indeed, I have seen it labelled as England on maps (albeit, only on simple maps that label just the countries with little or no further detail), and in informal speech this usage is fairly universal here. Yes, sometimes it is also called 'Britain', which is just as technically incorrect. Technically, Britain is an island (or, arguably, an archipelago) and does not include Northern Ireland, Gibraltar, or cetera, nor probably the Isle of Mann, yet people happily refer to the entire nation as Britain, or as England. (In the US, "Britain" and "England" are synonyms. Of the two, "England" is the more common term. Both are *way* more common than saying "the UK" much less "the United Kingdom" or anything longer.)
Calling the US "America" occasionally irritates people too (mostly Canadian pedants), but that doesn't really stop anyone from doing it.
I have *occasionally* noticed people saying "England" or "English" when they
mean the entire Commonwealth, but that is much less common. (One example
I specifically recall is the time my sister told me "Color doesn't
have a u in it. This isn't England." She knows perfectly well that they
spell it the same way in Australia as in Scotland or Wales or England proper.
As I said, however, this usage is not very common.)