For Scottish Nationalists, there has been one argument on this- Scotland is currently a member of the EU, and will remain so. And even if Scotland left the EU, then so would the rump state left, comprised of England, Wales and Northern Ireland.
We have been in a similar situation before. The United Kingdom of today is not the United Kingdom of 80 years ago. Well, it is, nnd it isn't.
So, here follows a (quite lengthy) look at history.
In December 1921, London and Dublin signed what is often called the Anglo-Irish Treaty, and the following March, the Westminster Parliament passed the Irish Free State (Agreement) Act 1922. There was a significant provision- it stated that when the legislation for ratifying the Treaty was passed, then Northern Ireland would have a right to opt out. In December, the Irish Free State Constitution Act 1922 was passed, and the devolved regions of Southern Ireland and Northern Ireland formed the Irish Free State- the day after this, the Parliament of Northern Ireland exercised its right to opt-out of the Irish Free State.
Under the Royal & Parliamentary Titles Act 1927 the United Kingdom of Great Britain and Ireland chamged its name to the United Kingdom of Great Britain and Northern Ireland.
The thing is, no-one says that one nation ceased to exist and two new nations took its place. If Scotland became independent, surely it would be like the Irish Free State situation- one devolved area of the United Kingdom leaves, and the bulk of the United Kingdom carries on.
As to the name of the rump nation, a bit of history- there used to be a sovereign nation called the Kingdom of England (or England for short), and this term included Wales. In July 1706, England and Scotland- which had had a common Head of State since March 1603 (when James VI, King of Scots, had become James I of England) signed a Treaty, and the English and Scottish Parliaments each passed an Act of Union the following year and in May 1707 the new nation of Great Britain was formed.
Ireland had its own Parliament, which, under the Crown In Ireland Act 1542 legislated that the English monarch would be Ireland's Head of State.
In the summer of 1800, the Parliaments of Ireland and Great Britain each passed an Act of Union, and on 1 January 1801, the first day of the 19th century and the day that Giuseppe Piazzi discovered the dwarf planet Ceres, the United Kingdom of Great Britain and Ireland was formed.
Looking at how Great Britain and then the United Kingdom were formed, a name could be the United Kingdom of England and Northern Ireland, drawing its name from the two Kingdoms (well, one Kingdom and part of an old Kingdom) that would form the remaining country.
In one part of the nation, such a name would not go down well! The Wales & Berwick Act 1746 made clear than any references to "England" in legislation (and this was applied retrospectively) would include Wales and Berwick-upon-Tweed. The Sunday Closing (Wales) Act 1881 was ground-breaking in that it applied only to Wales (by tha way, the legislation simply closed pubs on Sundays- from the title you might think that one couldn't visit Wales on a Sunday as it was closed).
Hidden away in the Welah Language Act 1967 is the crucial legislative change that references to England in future legislation did not include Wales. So, the post-1881 system whereby Wales was considered a part of England which might sometimes have special legislation applied to it was replaced by one where Wales was not considered part of England. Note that this legislation was passed the year after Gwynfor Evans won the Carmarthen by-election (July 1966) to become Plaid Cymru's first MP.
The Local Government Act 1972 (in a clause that curiously was repealed by the Local Government Act 1992) settles Monmouthshire's status by placing it firmly in Wales- although not everyone agrees, given that at the May 2011 elections to the Welsh Assembly, the English Democrats contested Monmouth.
So, with history lesson behind us, the logical name for the remaining nation would be "the United Kingdom of England, Wales and Northern Ireland", which would be shortened to "the United Kingdom", although most English people will shorten it to "England".
Three models are identified:
In the first model, Scotland secedes from the United Kingdom and forms a new state. In the third model, the United Kingdom ceases to exist and is replaced by two new states (in which case, the United Kingdom would have to re-apply for United Nations membership and perhaps have to wait its turn to be voted a temporary member of the Security Council, although it could argue that Russia simply took over the Union of Soviet Socialist Republic's place).
But, 1706 and all that. What I am thinking is that it would be the second model. England didn't occupy Scotland, or vice versa. Two independent nations had Parliaments that voted to unite. Now, some Scots might argue that the Scottish Parliament was bribed or blackmailed into voting for Union- but surely it is a case that two nations united voluntarily and then- if the Scots vote for independence- voluntarily separated. And therefore, the United Kingdom is in a unique position among EU member nations.
If this model is accepted- and we have history on our side to argue it- then Scotland becomes the EU's 29th member upon independence.
Now, either way, if Scotland does vote for independence, then the EU question has to be finally settled. The current deal is for the Scots to have just one question on the ballot paper, but surely there is space for a second referendum question alongside it, on a seperate ballot paper- If Scotland becomes an independent nation, do you believe it should be a member of the European Union?.
What if an independent Scotland wants to be part of the EU, but Brussels rules that it isn't and it has to join the queue? Well, I can see three options:
So, that third option could mean that the Parliament of an independent Scotland authorises British minister- after consultation with Scotland's Foreign Secretary- to represent them in Europe for the interim period, and conversely, British ministers make themselves available for questioning by Scotland's Parliament.