In this article, we propose a novel approach, called InPTC (Integrated Planning and Tube-Following Control), for prescribed-time collision-free navigation of wheeled mobile robots in a compact convex workspace cluttered with static, sufficiently separated, and convex obstacles. A path planner with prescribed-time convergence is presented based upon Bouligand's tangent cones and time scale transformation (TST) techniques, yielding a continuous vector field that can guide the robot from almost all initial positions in the free space to the designated goal at a prescribed time, while avoiding entering the obstacle regions augmented with safety margin. By leveraging barrier functions and TST, we further derive a tube-following controller to achieve robot trajectory tracking within a prescribed time less than the planner's settling time. This controller ensures the robot moves inside a predefined ``safe tube'' around the reference trajectory, where the tube radius is set to be less than the safety margin. Consequently, the robot will reach the goal location within a prescribed time while avoiding collision with any obstacles along the way. The proposed InPTC is implemented on a Mona robot operating in an arena cluttered with obstacles of various shapes. Experimental results demonstrate that InPTC not only generates smooth collision-free reference trajectories that converge to the goal location at the preassigned time of $250\,\rm s$ (i.e., the required task completion time), but also achieves tube-following trajectory tracking with tracking accuracy higher than $0.01\rm m$ after the preassigned time of $150\,\rm s$. This enables the robot to accomplish the navigation task within the required time of $250\,\rm s$.