Close to tears, milliner greatly upset (7)
I believe the answer is:
shatter
'greatly upset' is the definition.
(I've seen this before)
'close to tears milliner' is the wordplay.
'close to' suggests the final letters (the close/ending of).
'milliner' becomes 'hatter' (synonyms).
The last letter of 'tears' is 's'.
's'+'hatter'='SHATTER'
(Other definitions for shatter that I've seen before include "wreck" , "total" , "Break suddenly" , "Break into fragments" , "Break into many pieces" .)