British drive forwards and flourish (7)
I believe the answer is:
'flourish' is the definition.
(I've seen this in another clue)
'british drive forwards' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'drive' becomes 'urge' (I've seen this in another clue).
'forwards' becomes 'on' (synonyms).
'and' acts as a link.
(Other definitions for burgeon that I've seen before include "produce sprouts" , "Grow, rapidly increase" , "develop quickly" , "swell" , "bud" .)