allmusic lists two CDs that contain WHAT CAN YOU GIVE A NUDIST ON HIS BIRTHDAY?, both by Gracie Fields: "You Have to Laugh Don't You?" ASV/Living Era #5210, 1997; and "Our Gracie: 23 Favorites 1928-1947," ASV/Living Era #5259, 1998.

The composer was "A. Le Clerq"

Fortunately, one of the sound samples I listened to contained the "mystery line" marked with question marks above. I believe it should be:

"I've been in holy terror that I make an error."

There is also WHAT CAN YOU GET A NUDIST FOR HER BIRTHDAY? recorded by Bea Arthur on "Bea Arthur on Broadway: Just Between Friends," DRG #12993, 2002. Here's an excerpt:

...So what I want to know:
What can you get a nudist for her birthday?
What kind of birthday present can you buy?
Tell me, on the level, can I send her a muff?
[Followed by a long pause in which the audience laughs. Another forum says the following line is:
"I know it's true they wear a smile, but that can't be enough."]

A web page called Sounds of the Music Hall: A Discography lists an album called "Treading The Boards: Stars of Music Hall and Variety" which contains Leslie Sarony's recording of WHAT CAN YOU GIVE A NUDIST ON HER BIRTHDAY?