Former US singer not natural in nursery rhyme (3,4,4)
I believe the answer is:
old king cole
'nursery rhyme' is the definition.
(Old King Cole is a nursery rhyme)
'former us singer not natural' is the wordplay.
'former' becomes 'old' (both can mean previous).
'us singer' becomes 'Nat King Cole' (American jazz singer).
'not' suggests deleting specific letters.
'natural' becomes 'nat' (academic abbreviation).
'natkingcole' with 'nat' taken out is 'kingcole'.
'old'+'kingcole'='OLD KING COLE'
'in' acts as a link.
(Another definition for old king cole that I've seen is " Merry soul of rhyme".)