Cold call English luminary -- how embarrassing! (12)
I believe the answer is:
cringeworthy
'embarrassing' is the definition.
(cringeworthy can mean very embarrassing)
'cold call english luminary' is the wordplay.
'cold' becomes 'c' (eg on taps).
'call' becomes 'ring' (telephone call).
'english' becomes 'E' (abbreviation as in OED).
'luminary' becomes 'worthy' (both can mean an impressive person).
'c'+'ring'+'e'+'worthy'='CRINGEWORTHY'
'how' is the link.
(Other definitions for cringeworthy that I've seen before include "Severely embarrassing" , "Causing embarrassment" , "Acutely embarrassing" .)