Nigel's put out by Natasha's first in Cyrillic language (7)
I believe the answer is:
english
'cyrillic language' is the definition.
'english' can be an answer for 'language' (English language is an example). I'm not sure about the 'cyrillic' bit.
'nigel's put out by natasha's first' is the wordplay.
'put out' indicates anagramming the letters.
'by' means one lot of letters go next to another.
'natasha's first' becomes 'sh' (I can't justify this - if you can you should give a lot more credence to this answer).
'nigel' with letters rearranged gives 'engli'.
'engli'+'sh'='ENGLISH'
'in' is the link.
Can you help me to learn more?
(Other definitions for english that I've seen before include "Language spoken in his glen, strangely" , "An official language of the Republic of India" , "The nationality in his glen" , "to write these answers in?" , "Shingle (anag)" .)