High spirits of nephew, say, losing head (7)
I believe the answer is:
elation
'high spirits' is the definition.
('elation' can be a synonym of 'high spirits')
'nephew say losing head' is the wordplay.
'nephew say' becomes 'relation' (nephew is a kind of relation).
'losing head' means to remove the first letter.
'relation' with its first letter removed is 'ELATION'.
'of' is the link.
(Other definitions for elation that I've seen before include "exhibits successful pride" , "Feeling of joy and pride" , "Ecstasy -- toenail (anag)" , "Feeling of joy from toenail" , "Feeling of intense delight" .)