The most common and most likely answer to this crossword clue is the 13 letter word UNDERTATEMENT
I've seen this crossword clue in The Daily Telegraph
U
N
D
E
R
T
A
T
E
M
E
N
T
Buddy explains!
The straight part
We know that "That’s putting it mildly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that’s putting it mildly = undertatement.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.