IIRC, it was about four or five months. His first word was, 'what?' with the end of the word up as if asking a question. Why is that? Because every time he made a noise that we thought was a word, we'd say "what?" so that was his response, After that, it gradually escalated until a year in, we were hearing new things every week or even every few days. He lived with the radio or TV on most of the time, so when he wasn't hearing us talk,he was hearing English spoken pretty much all day long. We used to use a steno notebook, with each page for one letter of the alphabet, and we'd write the new word with the corresponding letter. As we gradually started to fill up the pages we gave up. He no longer learns as many new things, but when he goes through his reportoire at dusk or dawn, I occasionally hear him doing voices of people he hasn't seen in over 20 years. BTW, I think he's learning Polish from the people who talk outside our front window. But I can't be sure, because I don't speak Polish, so the random words just sound like gibberish to me.
The way I taught him new things that I wanted him to say, was simply to turn off all the noise, and just repeat the word to him while he was sitting on my shoulder. Other than that, he just chooses what he likes to say. He's particularly fond of the Pillsbury doughboy's 'ooo hooo' (like when someone poked him in the stomach). One of my favorites? "I'm a coooooool bird. I'm a coooooool bird. Are you a cool bird? I'm a cool bird". I still don't know why he knows to pronounce the word cool differently, other than maybe he considers the whole sentence one word, because the inflections are different.
Also, it helps to say whatever you're doing. Every time you open the door to the cage, perhaps say 'open the cage'. When you give him new water, say new water. If I accidentally forget to give him fresh water, sometimes he'll say "water".
So good luck. Keep talking to your birds!