The speed of fame -- not British (8)
I believe the answer is:
celerity
'the speed' is the definition.
(celerity is a kind of speed)
'fame not british' is the wordplay.
'fame' becomes 'celebrity' ('celebrity' can be a synonym of 'fame').
'not' suggests deleting specific letters.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'celebrity' with 'b' removed is 'CELERITY'.
'of' is the link.
(Other definitions for celerity that I've seen before include "Haste" , "Expedition" , "Swiftness of movement" , "Rapidity, speediness" , "Quickness" .)