How does the physiotherapist determine when I'm ready for stretching and strengthening exercises after my ankle sprain? When is the right time?

Varies. Remember you are dealing with experts- physiotherapist require years of training before they receive their degrees. The transition from symptomatic treatment to more active stretching is typically based on decreased swelling and pain. As symptoms further improve strengthening exercises are added. If at any point progression is difficult a return visit to physician is warranted.
Depends. On the severity of the ankle injury. Often the ability to start physiotherapy is based on a patient's ability to tolerate gentle manipulation and controlled motion. This is could be immediately for low grade sprains, several days for a moderate sprain and within a week for more severe sprains. Often a physician will help guide when a patient can start PT based on specific ankle protocols.