https://students-residents.aamc.org...le/mcat-testing-calendar-score-release-dates/
AMCAS app tends to open for editing on the first day of May (plus or minus a few days).
App tends to allow submission on the first day of June (plus or minus a few days). However, the first day a medical school will see your app (assuming the same process is followed this upcoming year as previous years) will likely be in the end of June, so as long as you are verified by then, you will be in the first batch of applicants.
If you wanted to know your score before the first day of possible submissions, the last possible date would likely by April 25, 2020 (and you'd get your score on May 27, 2020).
If you're fine with submitting a bit later as long as you can get verified before the end of June and be in the first batch, you can likely take your MCAT in May (likely before May 16, 2020) and get your score on June 16, 2020.
ALL OF THIS INFORMATION IS FOR AMCAS. I have nothing to contribute regarding AACOMAS dates and deadlines, as I know nothing about those.