The most common and most likely answer to this crossword clue is the 6 plus 10 letter word ZAPHOD BEEBLEBROX
I've seen this crossword clue in The Times
Z
A
P
H
O
D
B
E
E
B
L
E
B
R
O
X
Buddy explains!
The straight part
We know that "double-header on radio show" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case double-header on radio show = zaphod beeblebrox.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.