In 1603 King James VI of Scotland became King James I of England. ("England" in the Monarch's title included the Principality of Wales).
James succeeded Queen Elizabeth 1st (The Virgin Queen). He was the most likely to succeed Elizabeth as he was the great-great grandson of Henry VII of England (and he was a Protestant!).
James was a Monarch with two distinct realms, Scotland and England, each with a Parliament. He preferred to call himself King of Great Britain but in truth he was King of Scotland and King of England.
("Great Britain" is a geographical term for the land mass which comprises Wales, Scotland, and England. It has nothing to do with national "greatness").
The "Epistle Dedicatory" of the 1611 Authorised (King James) translation of the Bible (read it if you are interested to see how the translators "sucked up" to King James), styles him thus
TO THE MOST HIGH AND MIGHTY PRINCE
BY THE GRACE OF GOD
KING OF GREAT BRITAIN, FRANCE, AND IRELAND,
DEFENDER OF THE FAITH, &c.
The Translators of the Bible wish
Grace, Mercy, and Peace
through Jesus Christ our Lord
The claim to be King of France is a silly bit of historical nonsense. He was King of Ireland as a result of English colonialism, not because the Irish wanted him as Monarch.
The Monarchs from James VI and I until Anne (the last of the Stuarts) were Kings (they were all male) both of England and of Scotland. There was an English Parliament and a Scottish Parliament.
In 1706/1707 the two Kingdoms were merged - hence the United Kingdom - ( 1706 the English parliament voted in favour of union with Scotland, 1707 the Scottish parliament voted in favour of Union with England ).
The Parliaments were merged, and the new U.K. parliament was based in Westminster (not in Holyrood!).
Ireland was legally subordinate to the U.K. and was not formally a part of the Union until 1801, (remaining as a part of the Union until the partition of Ireland in 1922).
(In 1922 the Irish Free State was established - within the Commonwealth. Given the opportunity ,the majority Protestants in north east Ireland opted out of the Irish Free State, leading to the division of Ireland into the Republic of Ireland (which left the Commonwealth), and Northern Ireland - staunchly loyalist, unionist and anti-Catholic),
So Elizabeth II is Queen of the United Kingdom and Northern Ireland. She is also the Queen of Australia, of Canada, and of New Zealand.
Should the "yes" vote prevail in the Scottish referendum (Thursday Sep 18th 2014) Scotland will not become independent overnight. First there will be a protracted series of discussions, agreements, disagreement and protocols between the existing (semi-autonomous) Scottish Parliament at Holyrood, and the U.K. parliament in Westminster.
There would be no more United Kingdom, but there is no reason why Elizabeth II could not become Queen of England (with Wales) and Northern Ireland, and Queen of Scotland. After all she is already Queen of Canada, New Zealand and Australia.
PS Wales has never been a sovereign nation, it has never had a parliament, and it has never had a national Monarch. For better or worse it was simply absorbed into English rule and law. That's why modern devolution has created a Scottish Parliament but a Welsh Assembly.