It's probably not a song. It's likely that it's just a filigree or other nonsensical ad lib that Borge used as the framework for the joke. Don't page any attention to the score on the piano, that's just another prop.
He was playing allegro because he was laying an egg.
This song is "I Can't Remember Love" and is sung by an artist name Anna Hauss. She is on FB and soundcloud, but the song from the series isn't posted anywhere. I assume it was written for the show, or they now hold the rights if it was an original by Hauss.
Possibly you heard George Clinton/P-Funk's "Atomic Dog" (1982), which first introduced the "Bow wow wow" line. It also could have been Public Announcement's "D.O.G. in Me" (1998), which also used the line at the end of the track (and was sued by the P-Funk "Atomic Dog" copyright holders).
"Atomic Dog", ...
I listened to the clip in question. I'm no dictionary of blues tunes, but the tune being played was not familiar to me.
The clip was slightly less than two choruses of a basic 12-bar shuffle blues with a V-I turnaround. To me, the right-hand part being played sounded pretty ornamental and repetitive.
Therefore, I'm guessing that it was just someone comping ...