It's only a problem if you view 'British' as 'white'.   I know plenty of asians (i.e. as both as in the Pacific and India/Pakistan sense, etc) who are 'British' (or, more appropriately, Scottish) and perfectly well 'integrated' - understandable given that they are born in bred in the UK.  
If you want to equate nationality, or rather national culture, with a single religion or ethnic group, you're heading for trouble.  
NB(1): That is pretty much a bollocks statement.  By British, they probably mean 'foreigner hating white-pride Nazis who fight the ****'  (apologies for the language, but that's the kind of mindset these people have
EDIT; edited it out.  i'm not comfortable with writing any sort of racist epiphet, even if it's to illustrate the idiocy of these people
). 
 I don;t think they class people who disagree with them as being British.
NB(2): The BNP also supports, in particular, a policy of forced repatriation for asians.  They're basically based on a policy of pseudo-nazist racism.