I am confused why this doesn't translated as "we wouldn't be allowed to sing"? One of the other sentences reads "Kdokoli by směl odejít." and is translated as "Anyone would be allowed to leave"
It is accepted, or at least it is in the list of accepted translations. Maybe Duolingo has some problems, or you had some typo.
What is the difference between "Nesměli bychom zpívat" and "nesměli jsme zpívat"?
"Nesměli bychom" is a conditional clause "we wouldn't be allowed". "Nesměli jsme" is the past tense "we were not allowed/we have not been allowed/we had not been allowed".
Why not, "We were not allowed to sing?"
That is not a conditional, it is just a past tense. Nesměli jsme zpívat. See other comments for more.