The most common and most likely answer to this crossword clue is the 4 plus 3 plus 3 letter word MISS THE CUT
I've seen this crossword clue in Independent
M
I
S
S
T
H
E
C
U
T
Buddy explains!
The straight part
We know that "Fail to progress/being late for hairdressing appointment?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fail to progress/being late for hairdressing appointment? = miss the cut.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.