Although most of Ireland became an independent state in 1922, it was as a dominion of the British Empire known as the Irish Free State. This came to an end in 1937 following a referendum on a new constitution, but some powers were still reserved to the United Kingdom. The final vestiges of constitutional attachment to the United Kingdom were only brought to an end when the Irish passed the Republic of Ireland Act, 1948.
In response to this, the United Kingdom Government passed the Ireland Act, 1949. Section 2(1) of this Act is particularly interesting:this page) and remains in force to this day.
That makes the repeated assertions by several unionist politicians about the Scots suddenly becoming "foreigners" if Scotland becomes independent—including this by David Cameron and these by Ed Miliband and other prominent Labour and LibDem politicians—not only hollow and ridiculous, but dreadfully misinformed.
There is absolutely no reason why the remainder of the UK should treat an independent Scotland and its people any differently from the way the (somewhat larger) remainder of the UK treated the Republic of Ireland and its people in 1949.