Lucy shows up at around 12 minutes in. Bugs Bunny also makes an appearance.
https://free-classic-radio-shows.com/Comedy/Abbott-and-Costello/1943-11-18-Abbott-and-Costello-w-Lucille-Ball/index.php

I have that episode! Costello wants to go out with Connie Haines(She was the show's singer),but only if Costello buys Nylon stockings for her. He tries to get some, but Lucille Ball gets them first. So Abbott & Costello go to her house to try to get the stockings back. This was aired in 1943 during WW2 when there were many shortages of material which were used to help the solders fight the ememy.