We study the quantum Hall effect (QHE) in the three-dimensional topological insulator HgTe, which features topological Dirac-type surface states in a bulk gap opened by strain. Despite the coexistence of multiple carrier subsystems, the system exhibits perfectly quantized Hall plateaus at high magnetic fields. Here we study the system using three different experimental techniques: Transport experiments, capacitance measurements including the quantum capacitance, and current distribution measurements using electrostatically sensitive scanning probe microscopy. Our key finding is that at sufficiently high magnetic fields, the different electronic subsystems merge into one, and the current in a quantum Hall plateau is distributed across the entire width of the Hall bar device.