Nasty bug from priest keeping company (1,4)
I believe the answer is:
e coli
'nasty bug' is the definition.
(I've seen this before)
'priest keeping company' is the wordplay.
'priest' becomes 'eli' (I've seen this before).
'keeping' means one lot of letters goes inside another.
'company' becomes 'co' (abbreviation).
'eli' enclosing 'co' is 'E COLI'.
'from' is the link.
(Other definitions for e coli that I've seen before include "Gut bugs" , "Common species of bacterium" , "cause of ill-feeling" , "Bacterium found in the gut" , "An infection found in food and water" .)