The most common and most likely answer to this crossword clue is the 5 letter word APIAN
I've seen this crossword clue in The Times
A
P
I
A
N
Buddy explains!

The straight part
We know that "Like Flight of the Bumblebee" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case like flight of the bumblebee = apian.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 APIAN mean?
APIAN noun- (very rare) A bee.
- Relating to bees.