One of the cats, each with the cuckoo (7)
I believe the answer is:
cheetah
'one of the cats' is the definition.
(cheetahs are big cats)
'each with the cuckoo' is the wordplay.
'with' means one lot of letters go next to another.
'cuckoo' indicates anagramming the letters.
'each' after 'the' is 'theeach'.
'theeach' with letters rearranged gives 'CHEETAH'.
(Other definitions for cheetah that I've seen before include "Fast cat - the ache (anag)" , "Feline, the fastest mammal" , "Swift large wild cat with spotted coat" , "Fast predator" , "Long-legged spotted big cat" .)