technically once you're able to run around a course together (probably could do that within a few months) you can compete. But you won't be
competitive for much much longer (a year or more) so I guess it depends on what your goal is.
You'll find many of the good trainers will wait until their dog is REALLY going before ever entering a competition, it could take a year or so of training, but when they enter a competition, they are highly likely to place and win up to the next level pretty fast. Generally those trainers have already done some agility though, and know the deal, so it isn't worth the entry fee and petrol to go to a show just to play about! Whereas newer/first time trainers are more keen to just go out there ASAP and say they are competing with the big guys! lol!
So, if you are only interested in getting out ASAP to play on a course in a 'real' show, then by all means you'd be able to do that pretty quickly. If you want to go out there with a competitive dog who will run fast and accurate, then the sky is the limit as to how long it will take of training to get there.