The most common and most likely answer to this crossword clue is the 6 letter word BIRTHS
I've seen this crossword clue in The Daily Telegraph
B
I
R
T
H
S
More clues leading to the result BIRTHS
Buddy explains!

The straight part
We know that "deliveries" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case deliveries = births.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 BIRTHS mean?
BIRTHS noun- The process of childbearing; the beginning of life.
- An instance of childbirth.
- A beginning or start; a point of origin.
- The circumstances of one's background, ancestry, or upbringing.
- That which is born.