Some thoughts and points:
As a 'not fully independent' nation, as the SNP is proposing, would Scotland be able to remain in the EU with us? Or would they have to apply as their own country? Aside from the economic requirements, one of the rules of membership is the adoption of the Euro. I can't see that being attractive.
Forget the flag, a more pressing issue would be the name - could we still be able to title ourselves as 'The United Kingdom of Great Britain and Northern Ireland'? Great Britain is the consituent parts of England, Wales and Scotland, the name that came about because of the Union of England and Scotland.
But there is no reason our flag would need to change just because Scotland left.
Firstly, it features in so many other flags around the world, those of the Commonwealth nations and British soverign territories that to continue that bond and recognition of those states it should remain the same.
Secondly it is also iconic and recognisable - we've had this version of it since 1801! But for the sake of argument, the white of the St Andrews cross could switch to representing the white of the St Patricks flag, like the red St Georges one has the white outline. The blue could be both a historical nod to our island components and represent the sea. And let's be honest, whatever alternative design might be come up with, chances are the general public will agree they prefer the current one better.
Thirdly:
Hawaii's state flag - notice that British Flag in the corner there? THEY'VE never changed theirs!
If the flag were to change, we could be technical about it and represent Wales with use of the Flag of St David, matching the use of the flags of patron saints of the other members on the national flag. (Scotland also has the Soverign's flag of the red lion on gold). Which could make our flag this instead, if we so wished:
http://prntscr.com/2nr630
(sorry it's a hyperlink, photobucket is down...)
As a side note, the Welsh flag of the dragon was only made the official national flag of Wales in 1959.