The most common and most likely answer to this crossword clue is the 4 plus 4 letter word TAKE ROOT
I've seen this crossword clue in The Guardian
T
A
K
E
R
O
O
T
More clues leading to the result TAKE ROOT
Buddy explains!
The straight part
We know that "Become established down under" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case become established down under = take root.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.