The traditional hand rehabilitation robots typically employ multiple actuators to independently control finger motions with the high system complexity and cost, which limits the lightweight design and practical deployment. Although the underactuated mechanisms can reduce the number of actuators, their cable length adjustment capability is structurally constrained, which limits the adaptability to different hand sizes and initial postures and also degrades the motion response speed. To address these issues, a slack-adjustable underactuated cable-driven actuator is proposed for hand rehabilitation robots, where the actuator employs a dual-pulley differential mechanism to separate the driving cable from the finger cables. Here a knob-based adjustment mechanism is introduced to independently regulate the initial slack of each finger cable without relying on the motor operation, which effectively reduces the initial dead zones and improves the motion response performance. Meanwhile, the finger cables are arranged in a differential mechanism within the dual-pulley system, enabling the automatic redistribution of cable length and tension according to different finger blockage conditions during grasping, thereby realizing the adaptive grasping. The system is driven by a single motor and integrates gear transmission with the dual-pulley differential mechanism to realize the coordinated control of thumb, index finger, and middle finger. Experimental results show that the proposed slack adjustment mechanism reduces the average system response time by about 91.5% with an initial slack of approximately 10 mm, which significantly enhances the motion responsiveness. Additionally the stable adaptive grasping performance is maintained under the various finger-blocking scenarios. The measured fingertip forces of thumb, index finger, and middle finger reach 8.85, 8.29, and 7.84 N, respectively, which meet the force assistance requirements for the hand rehabilitation training.