If it's a slam-dunk "documentary" case (where a mere document shows that the first marriage was not valid) it could be very quick, whereas if it's a contested annulment which requires tracking down the respondent, translators, cross-examinations, etc. it could be a long time coming, and a favorable ruling is by no means certain.
The pope wasn't willing to compromise with King Henry VIII on this, even though it cost him England and pret-near the whole Anglosphere for the next 500 years.
Hate to break the bad news to you, but there's a new pope in town now.