How long should I expect it to take to notice progress in my flexibility?

Your body has a biological time for all the activities you do. So, 21 days is the time which you develop a habit or see your body changing dramatically.

It depends on how much you are used to stretching. In my personal case I stopped stretching for years. Then I started again and after about a week of stretching for 7 minutes every day, I did notice a big improvement in my mobility.