When the wizards reside in the southeastern portion (and also east of Wales) of the British Isles they are English wizards. The term "England" is used in the book series mostly when there is a specific place in England:
“ ‘Upon the signature of the International Statute of Secrecy in 1689, wizards went into hiding for good. It was natural, perhaps, that they formed their own small communities within a community. Many small villages and hamlets attracted several magical families, who banded together for mutual support and protection. The villages of Tinworth in Cornwall, Upper Flagley in Yorkshire, and Ottery St. Catchpole on the south coast of England were notable homes to knots of Wizarding families who lived alongside tolerant and sometimes Confunded Muggles.
For everything else, JKR uses "Britain." I do not think the wizarding community cares about the constructs like the United Kingdom, which involves Northern Ireland. But I do think that Britain at the very least means Great Britain. How did I get here? This quote:
“Attendance is now compulsory for every young witch and wizard,” he replied. “That was announced yesterday. It’s a change, because it was never obligatory before. Of course, nearly every witch and wizard in Britain has been educated at Hogwarts, but their parents had the right to teach them at home or send them abroad if they preferred. This way, Voldemort will have the whole Wizarding population under his eye from a young age.
About the above: Remember that Hogwarts is in Scotland and thus is definitely included in "Britain."
Here's another quote:
“Level seven, Department of Magical Games and Sports, incorporating the British and Irish Quidditch League Headquarters, Official Gobstones Club, and Ludicrous Patents Office.”
Notice it wasn't Welsh, English, Scottish, and Irish Quidditch League Headquarters? This may also mean that Ireland (whether divided or whole) falls under the British Ministry of Magic in SOME respect. Though, to what extent, it doesn't say in canon.
I conclude with a none-canon, albeit noteworthy, reference: JK Rowling, who is from Scotland, was a champion for the NO vote or Better Together campaign.