The most common and most likely answer to this crossword clue is the 10 letter word CHILDBIRTH
I've seen this crossword clue in Independent
C
H
I
L
D
B
I
R
T
H
Buddy explains!

The straight part
We know that "delivery" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case delivery = childbirth.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.
What does CHILDBIRTH mean?
CHILDBIRTH noun- The fact or action of giving birth to a child, as the culmination of pregnancy.