The most common and most likely answer to this crossword clue is the 9 letter word TOPIARIST
I've seen this crossword clue in The Times
T
O
P
I
A
R
I
S
T
Buddy explains!
The straight part
We know that "Stylist working on Bush" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case stylist working on bush = topiarist.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.