It’s called Nothing Like Broadway!, and it actually takes place in a little theater. Milo, the sweet, schlubby lighting booth guy (who also runs the actual lights for the actual show) dreams of singing onstage with his friend Annabelle, a wealthy patron of the arts. The story is what happens when two rival super-spies— Bixby, a dashing if bumbling British agent, and Kishka, a lascivious Russian woman— both decide to hide out in the tiny theater, going undercover as cabaret singers, not knowing the other is doing the same. Farce ensues. The theater ends up being the center of a global conflict, with a nuclear missile pointed at the theater and at the audience (who are the actual audience). It all works out in the end.
Here's the opening number, "This is the Start of Something".
Here is the title song, "Nothing Like Broadway".
And this is Bixby's song, "The World's Most Famous Spy".
You can hear even more songs from the show here.