Caught one at tea party with Queen that’s a fast talker (9)
I believe the answer is:
chatterer
'a fast talker' is the definition.
'caught one at tea party with queen' is the wordplay.
'caught' becomes 'c' (cricket abbreviation).
'one at tea party' becomes 'hatter' (character at a tea party in Alice in Wonderland).
'with' means one lot of letters go next to another.
'queen' becomes 'er' (abbreviation for Elizabeth Regina).
'c'+'hatter'+'er'='CHATTERER'
'that's' acts as a link.
(Other definitions for chatterer that I've seen before include "babbler" , "Gossip" , "One talking idly or rapidly" .)