Conservative worn out following public complaint (10)
'complaint' is the definition.
(medical complaint)
'conservative worn out following public' is the wordplay.
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'worn out' becomes 'old'.
'following' means one lot of letters go next to another.
'public' becomes 'common' ('common' can be similar in meaning to 'public'**).
'c'+'old'='cold'
'cold' after 'common' is 'COMMON COLD'.
(Other definitions for common cold that I've seen before include "Regular winter illness" , "Mild viral infection" .)