They're wrong about British in consequence (7)
I believe the answer is:
thereby
'in consequence' is the definition.
(I've seen this before)
'they're wrong about british' is the wordplay.
'wrong' is an anagram indicator.
'about' means one lot of letters goes inside another.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'theyre' anagrammed gives 'therey'.
'therey' placed around 'b' is 'THEREBY'.
(Other definitions for thereby that I've seen before include "By that means" , "As a result of that" , "about that amount" , "In consequence" , "For that reason" .)