I had all my paperwork complete and swore in October, 2001. It took me about 10 months to get approved. As soon as I got the approval they sent me to Brooks three weeks later. After getting through medical flight screening, the AMS dates came about 2 weeks later. The UPT dates also came at this time, and they tell you what plane you will fly after the T-37, or T-6.
There are 4 other candidates that started around the same time and three of us now have AMS and UPT dates. The wait really bites but it will all be worth it.
From what I have heard, the time it takes to get approved varies. I know guys who have been approved in six months and some that took over a year. The key for me was to keep bugging anybody who had any say in getting my paperwork to the right people. I think making several phone calls and visits to these people really helps.