I sprained my sacro-iliac (SI) joint about 8 years ago. I'm sorry to say it still bothers me occasionally to this day, but I didn't rehab it nearly as well as you are doing (I was young, dumb, and had bad health insurance, so I only did 2 weeks of PT and went back to playing Ultimate and soccer).
IMO, if you follow through your entire rehab, it will probably take 2-3 months to heal. However, like most sprains, the muscle will probably be weak and/or more susceptible to injury for years.
As for it affecting cycling in the future, I've never had cycling make my SI flare up. However, I ride a mountain bike, which keeps me more upright. I've had several doctors tell me (and personal experience confirms) that bending forward at the waist is really bad for sacrum injuries. A recumbent will probably be perfect; riding a road bike in an aero position *might* cause a flare-up (and I certainly wouldn't get on a road bike if my SI was bothering me).
I hope your experience is much better than mine.
