Why no Scottish nor Northern Irish players in Team GB

Since a few months ago, I've been wondering how to form the football GB Team, how many players from each country would be representing the whole kingdom.
Of course, the Olympic Games have already started. And they have started in London, United Kingdom. The very same United Kingdom formed by not one nor two nor three countries but by four: England, Scotland, Wales and Northern Ireland.
And, of course, there are lots of different sports to be played in the Olympics. And I understand that, in some sports, olympians from one UK country are better than others.
I understand that, for example, the Scottish tennis player Andy Murray (ranked 4th in the ATP) is a much better player than any other British tennis player. And I understand that there can be Englishmen or Welshmen or Northern Irish athletes who are better than other Britishmen in their respective disciplines. But most of the sports in the Olympics are single players disciplines. Or, they are for just a few.
So, there is my question:
Why in a sport like football which requires not one nor two nor four but EIGHTEEN men per squad the coach picked just English and Welshmen?
Does this represent the whole country? I don't think so.
Could they have picked players from Scotland and Northern Ireland? Of course.
Could they have picked, at least, just ONE Scotsman and ONE Northern Irishman? Of course.
Scottish and Northern Irish good football players weren't good enough to be selected for the GB team? Northern Irish maybe. Scottish, I don't think so.
Does Stuart Pierce have any problem with Scotland and/or Northern Ireland? Who knows.
So why the hell there are no players representing the 6 million Scottish nor the 1.8 million Northern Irish?
Couldn't they take AT LEAST ONE PLAYER from each country?

Well, since I DO think that the GB Team should represent everyone in the United Kingdom of Great Britain and Northern Ireland, I picked my GB Team where there are not one but at least two players from each country, plus the chance of playing in the Olympics for the experienced David Beckham, Ryan Giggs and David James who form the over-23-year-old trio.

01) David James (GK-ENG-42)
18) Jack Butland (GK-ENG-19)
02) Kyle Walker (DF-ENG-22)
04) Phil Jones (DF-ENG-20)
05) Ryan McGivern (DF-NIR-22)
06) Craig Cathcart (DF-NIR-23)
12) Chris Smalling (DF-ENG-22)
15) Danny Wilson (DF-SCO-20)
03) Gareth Bale (MF-WAL-23)
07) David Beckham (MF-ENG-37)
08) Barry Bannan (MF-SCO-22)
11) Ryan Giggs (MF-WAL-38)
16) Aaron Ramsey (MF-WAL-21)
17) Tom Cleverley (MF-ENG-22)
09) Danny Welbeck (FW-ENG-21)
10) Daniel Sturridge (FW-ENG-22)
14) Theo Walcott (FW-ENG-23)

Note: I would have picked Andy Carroll (FW-ENG-23) but he has been injured.