The most common and most likely answer to this crossword clue is the 3 plus 4 letter word AGA SAGA
I've seen this crossword clue in The Daily Telegraph
A
G
A
S
A
G
A
More clues leading to the result AGA SAGA
Buddy explains!

The straight part
We know that "Warming tale" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case warming tale = aga saga.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.