Julie Andrews sings it on her album of the same title. My LP is still boxed after our move but I have transcribed a few of the items to a cassette also not immediately at hand. I also have sheet music from my wife's collection which I haven't completely sorted yet. Lyrics are in Frank Shay's My Pious Friends and Drunken Companions with the note "to be half-recited, half sung to any melody you can adapt to the words". Pious Friends lyrics, below, are a little different from what I remember of Julie Andrews'. Shay notes "as sung by Lawrence Grant and W.D.Smith".

DON'T GO IN THEM LIONS CAGE TONIGHT, MOTHER
(as sung by Lawrence Grant and W.D. Smith)

A lady once she had a lovely daughter,
The lady was an actress on the stage.
She traveled with a troupe of awful lions,
And each night she went into them lions' cage.
One night the daughter had a premonition
That everything that night would not be right,
And so she told her mother in the kitchen,
"Oh, don't go near them lions' cage tonight!"

CHORUS:
"Oh don't go near them lions' cage,
Dear mother, dear, tonight.
Them lions am ferocious and might bite!
And when they get them angry fits
They'll chew you into little bits.
Oh, don't go near them lions' cage tonight".

The lady laughed "ha ha!" she did not heed the warning
That unto her her daughter she did give.
"Oh, no," she cried, "I do not fear them lions:
Not one of them could make me cease to live."
She went into that cage of awful lions,
Them lions were ferocious as could be.
"Alas," she cried, as one strode up and bit her,
"I now recall what daughter said to me."

CHORUS

"Oh, who will save my mother?" cried the daughter.
"By lions she is being bit and et!"
"I will," replied a young man in the gallery;
"I'll save your mother from them brutes, you bet!"
He went into that cage of awful lions,
Of lion biting she was almost dead.
"Here is you ma," he said to her and kissed her;
For he the daughter loved and soon did wed.

Source: My Pious Friends and Drunken Companions, collected by Frank Shay and illustrated by John Held, Jr.
Macaulay Co., New York, 1927. pp 189-190