First remember everyone who was born in Britain, did not have choice in that matter so showing blind loyalty to Britain because of something you did not choose does not make sense.
However people do choose what religion they are so it makes sense to have a greater loyalty to the religion then the country you were accidently born in.
Having said this I would fight for Britain if the cause was just i.e fighting islamic state, however I would rather be imprisoned then fight with Britain in an unjust war.