How can consistent minimal exercise over long periods change your body?
It depends on a whole lot of factors. If your existing body weight is just a few kilos/pounds away from your ideal body weight, 15 to 20 min of moderate intensity exercise coupled with a decent diet can yield results in 4- 6 weeks.
Everything depends on your goal(fat loss, lean look, bulking, endurance building) and if your existing lifestyle can support that goal. Once you pick a specific goal, you will have a better idea to execute your plan.