I presume "exam hall" is a standard expression, proesumably American.
I presume that you use more than one. SO I add an "s" to it.
I wish I had a time machine that could store the time that I lose by singing silently, the time that passes simply doing nothing in exam halls because I can't answer and don't want to submit my test paper earlier than others for the fear of losing prestige. And when I lack time, I could use that stored time.
"passing" did not seem to work so well there: not incorrect, just not comfortable.
I changed "couildn't" to "can't." to agree.with "singing"
I simplified "the problem of lack of time" - not incorrect, but longer than necessary.