I took 5 months of lessons before my first competition, but the first 3 months are social dancing lessons, since that's what everyone does in that studio. Then I moved, and started taking lessons with an actively competing pro for another 2 months before my first comp. We won standard, and got second in smooth in the newcomer category (collegiate am/am). I just had another cmpeition (this is 9 months after first lesson, but 6 months after dancing competitively) with my am partner (collegiate), and we won bronze smooth, and got 4th in bronze standard.
My first pro/am competition is in April. By then, I will have been dancing with my standard teacher for 7 months, and my latin teacher for 5 months.
I think people start competeping rather early in the collegiate world? I don't know what the average wait before competition is in the pro/am world.