There is an article on the Ministers of Magic on Pottermore that states following:
All matters relating to the magical community in Britain are managed
solely by the Minister for Magic, and he has sole jurisdiction over
So the question would be: what is Britain?
According to wikipedia Britain may refer to an island or a country:
The term Britain is a linguistic descendant (reflex) of one of the
oldest known names for Great Britain, an island off the north-western
coast of continental Europe.
From the same article:
The term Britain is widely used as a common name for the sovereign
state of the United Kingdom, or UK for short. The United Kingdom
includes three countries on the largest island, which can be called
the island of Britain or Great Britain: these are England, Scotland
and Wales. However the United Kingdom also includes Northern Ireland
on the neighbouring island of Ireland, the remainder of which is not
part of the United Kingdom. England is not synonymous with Britain,
Great Britain, or United Kingdom.
Nevertheless at some point in time "Britain" was used as a synonym for the whole United Kingdom of Great Britain and Ireland which from 1801 to 1922 included also the current Republic of Ireland. It is unknown whether the wizard society also accepted this independence.
Ireland is not included in Britain but a separate island
Ireland (Listeni/ˈaɪərlənd/; Irish: Éire [ˈeːɾʲə] ( listen);
Ulster-Scots: Airlann [ˈɑːrlən]) is an island in the North Atlantic
separated from Great Britain to its east by the North Channel, the
Irish Sea, and St George's Channel.
and most of it is an independent country an independent country:
Ireland (Listeni/ˈaɪərlənd/; Irish: Éire [ˈeːɾʲə] ( listen)), also
known as the Republic of Ireland (Poblacht na hÉireann), is a
sovereign state in north-western Europe occupying about five-sixths of
the island of Ireland.
According to all this Ireland is an independent country also in the magical world UNLESS the state of the magical government is "frozen" in the state before the Irish independence in 1922 but after 1801.