The most common and most likely answer to this crossword clue is the 7 plus 8 letter word EXPRESS DELIVERY
I've seen this crossword clue in The Guardian
E
X
P
R
E
S
S
D
E
L
I
V
E
R
Y
Buddy explains!
The straight part
We know that "speedy service" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case speedy service = express delivery.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.