Can someone tell me how long will it take to see progress with insanity workouts?

Quickly: Through the 60 day program, metabolic parameters like weight, body fat content, blood pressure may take a while to come down but other changes like elevated mood, endurance, flexibility will seem to come within just a few days of doing it.

Quite variable: As it depends on what type of progress you are referring to, what your baseline fitness level is and how you approach(particularly intensity) this program in conjunction with your eating habits. If you are referring to weight loss look on the order of weeks(at least one).