Some of the challenges that need to be overcome include creating a reliable and efficient propulsion system, protecting against cosmic rays and developing materials resistant to the space environment. Furthermore, the safety and reliability of the system are critical aspects that need to be ensured before the space elevator can be used for human transportation.
While it's difficult to predict exactly when space elevators will be ready, it's possible that we'll see the first generation of these systems in use in space within the next 10 to 20 years. However, it is important to remember that technology is developing and this can be affected by many factors, including funding, technological advances and changes in space policies.