Pet has no time ultimately for teacher (5)
I believe the answer is:
rabbi
'for teacher' is the definition.
(Jewish teacher)
'pet has no time' is the wordplay.
'pet' becomes 'rabbit' (a rabbit might be kept as a pet).
'has no' is a deletion indicator.
'time' becomes 't' (abbreviation).
'rabbit' with 't' taken away is 'RABBI'.
'ultimately' is the link.
I am not very happy about this link. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for rabbi that I've seen before include "Spiritual leader of a Jewish congregation" , "Leader of worshippers" , "Legal expert" , "A Jewish clergyman" , "A Jewish priest" .)