I find it hard to believe you have no time, unless you are in med school. Even still, I have 2 friends that went through med school and still trained. I know working mothers with 2 babies a piece that still manage to train and take care of their famies.
Sorry for the bluntness, I just have a hard time dealing with people who don't make the time to take care of themselves. What's life without health?
Anyhow, maybe you will prove me wrong and have some crazy, insane schedule that I have never seen before.... if that is the case, I REALLY feel for you.
And to answer your question..... the minimum amount of exercise to lose weight. None.
Weight loss is completely about energy balance. Eat less then you expend and you will lose weight over time. That simple.
Optimal without cardio and resistance training? Absolutely not.