The most common and most likely answer to this crossword clue is the 2 plus 3 plus 4 letter word ON THE HOUR
I've seen this crossword clue in The Times
O
N
T
H
E
H
O
U
R
More clues leading to the result ON THE HOUR
Buddy explains!
The straight part
We know that "Punctually, like Barnardo?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case punctually, like barnardo? = on the hour.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.