The most common and most likely answer to this crossword clue is the 10 letter word UN-BRITISH
I've seen this crossword clue in The Guardian
U
N
-
B
R
I
T
I
S
H
Buddy explains!
The straight part
We know that "not befitting these isles" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case not befitting these isles = un-british.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.