The most common and most likely answer to this crossword clue is the 4 letter word GIGI
I've seen this crossword clue in The Daily Telegraph
G
I
G
I
More clues leading to the result GIGI
- Italian tenor losing line in musical
- Musical film soldier’s seen twice
- Colette’s soldiers
- Show privates one after performance
- Musical performance beginning to impress
- Musical soldiers
- Film produced by two US soldiers
- Musical performance one must follow
- Performance of rock band on one old musical
- Modern concert followed by a 50s film
Buddy explains!

The straight part
We know that "old musical" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case old musical = gigi.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.