Robust transmission in topological insulators makes it possible to steer waves without attenuation along bent paths within imperfectly fabricated photonic devices. But the absence of reflection does not guarantee the fidelity of pulsed transmission which is essential for core photonic functionalities. Pulse transmission is disrupted by localized modes in the bulk of topological insulators which coexist with the continuum edge mode and are pushed deeper into the band gap with increasing disorder. Here we show in simulations of the Haldane model that pulse propagation in disordered topological insulators is robust throughout the central portion of the band gap where localized modes do not arise. Since transmission is robust in topological insulators, the essential field variable is the phase of the transmitted field, or, equivalently, its spectral derivative, which is the transmission time. Except near resonances with bulk localized modes that couple the upper and lower edges of a topological insulator, the transmission time in a topological insulator is proportional to the density of states and to the energy excited within the sample. The variance of the transmission time at the band edge for a random ensemble with moderate disorder is dominated by fluctuations at resonances with localized states, and initially scales quadratically. When modes are absent, such as in the center of the band gap, the transmission time self-averages and its variance scales linearly. This leads to significant sample-to-sample fluctuations in the transmission time. However, because the transmission time is the sum of contributions from the continuum edge mode, which stretches across the band gap, and far-off-resonance modes of the passband and band edge, there are no sharp features in the spectrum of transmission time in the center of the band gap. Instead of creating obstacles to be circumvented with inevitable increase of the transmission time in the band edge mode, the transmission time of the band edge mode is decreased by disorder. As a result, ultrashort, broadband pulses are faithfully transmitted in the center of the band gap of topological insulators with moderate disorder and bent paths. This allows for robust signal propagation in complex topological metawaveguides for applications in high-speed optoelectronics and telecommunications.