The most common and most likely answer to this crossword clue is the 7 plus 5 letter word CURLING IRONS
I've seen this crossword clue in Financial Times
C
U
R
L
I
N
G
I
R
O
N
S
Buddy explains!
The straight part
We know that "Heating appliance used in Barnet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case heating appliance used in barnet = curling irons.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.