It just seems odd to me that the names under consideration are things like "Indian-British", or "Asian-British", neither of which exactly roll off the tounge (and are likely to be shortened to Indian or Asian in common usage anyway).
It seems to be blindly following the American example (Italian-American, Irish-American) without considering the linguistic ramifications.
American starts will a soft vowel and as such is suited to being the second word in a joined phrase. British starts with a very hard consonant group and is more suitable as the first word in the construct.
Indian-British vs. British-Indian*
Asian-British vs. British-Asian
Pakistani-British vs. British-Pakistani
All of which ignores the silly idea that attitudes will change simply because the words have changed - but if they're going to do this, at least do it right.
* I can't think of a single word with 'nbr' in the middle - but I can think of plenty with 'shi'.