The new sentence doesn't work. It doesn't say who is going to learn about the result.Just to clear things up.
After taking the exam, it takes three months to learn about the result. :tick:
What about my next sentence? Would this work?
After taking the exam, three months will elapse to learn about the result.
Don't worry about annoying us. Answering questions about English is why we're here.
Even if I annoy you, I'll try again.
Would the following sentence be correct?
After taking the exam, about 3 months will pass before you learn about the result.