Wheeled mobile manipulators operating in indoor unstructured environments, which are subject to nonholonomic constraints such as a minimum turning radius and the inability to perform instantaneous lateral translation, face significant challenges in whole-body motion planning. Existing decoupled approaches often result in fragmented final paths due to the disconnected planning between the base and the manipulator. Meanwhile, high-dimensional sampling methods suffer from low computational efficiency and poor realtime performance. To address these problems, an innovative hierarchical path planning method is proposed. The method first employs an improved bidirectional dual-tree rapidly-exploring random tree (RRT) algorithm. This algorithm directly incorporates a nonholonomic constraint model and simultaneously explores both forward and backward solution spaces to eliminate redundant computation, thereby generating a collision-free base path embedded with spatiotemporal information. Subsequently, for each temporal node along this base path, a set of feasible manipulator configurations is defined, all satisfying conditions including joint limits and static collision avoidance. Using these sets, a temporal-node-constrained RRT algorithm generates a manipulator path that is strictly synchronized with the base motion. Finally, the complete whole-body motion path is synthesized by aligning the base and manipulator waypoints through index matching. Both ROS simulations and physical experiments show that the algorithm proposed decouples the high-dimensional planning problem, enabling coordinated motion of the mobile base and manipulator in complex constrained environments while generating collision-free trajectories throughout the entire motion in real time. The experimental results indicate that the algorithm has significant advantages in planning time, path length, and smoothness. Compared with the RRT*-Connect algorithm, in complex scenarios, the proposed approach reduces planning time by 29.16% on average, decreases path length by 16.68%, and improves path smoothness by 56.13%.