Just to clear something up, you can not really draw comparisons between the American States and the various "nations" within the UK. Largely because of the thousands of years of warfare stretching back to Roman times between the British states. Scotland, N.Ireland and Wales also have their own languages, and cultures. Then in 1707 there was the act of the Union between England and Scotland, and Britain was formed. (Wales was already conquered by Edward III,(remember him from Braveheart?) and there was much unrest in Ireland until very recently.
The trauma of many centuries of warfare have been gradually healing since then, but how much powers the various states have is murky territory. Scotland Wales and N.Ireland where given more power in 1997, Scotland and Wales seem to have control over their own education, whilst receive money from England.It's all very confusing, and you and I are not the first and last people to be confused by the politics of the British Isles- to much history can be a bad thing XD
I just think about it like this- I am English national, I am also a British national, as well as an EU citizen.