At last, yesterday I listened to this radio show, although I did not do it before because I was busy.
"Did not" is a bit more direct. It implies that you recently had a chance to do something, and either were unable to, or chose not to.
At last, yesterday I listened to this radio show, although I had not done it before because I was busy.
"Had not" is a bit "loose." It implies that you should have done it in the past, but you have not done it up to this point.
However
I would rewrite this sentence if it was mine.
At last, yesterday I listened to the radio show
The first part rolls off the tongue uncomfortably. I would shorten it and remove the comma.
I finally listened to the radio show yesterday
, although I did not do it before because I was busy.
, although I had not done it before because I was busy.
I would also rewrite the second part, although that is mostly personal preference.
; I did not do it before because I was busy.
; I had not done it before because I was busy.
My final sentences (personal preference):
I finally listened to the radio show yesterday; I did not do it before because I was busy.
I finally listened to the radio show yesterday; I had not done it before because I was busy.