This paper investigates single-photon scattering and reflection nonreciprocity in a waveguide-QED system composed of a topological giant atom coupled to a one-dimensional infinite waveguide. The giant atom is formed by a finite Su-Schrieffer-Heeger (SSH) atomic chain, with only its first and last atoms coupled to the waveguide at two spatially separated points. In this configuration, the topological eigenmodes of the SSH chain and the accumulated propagation phase between the two coupling points jointly provide tunable degrees of freedom for manipulating single-photon transport. Using a real-space approach, we derive analytical expressions for the scattering amplitudes of a single photon incident from either the left or the right side of the waveguide. Based on these results, we systematically analyze the influence of the SSH topological phase, the accumulated phase, the number of atoms, atomic dissipation, and disorder on the scattering spectra.

The results reveal that the reflection of resonant photons can be continuously tuned by changing the ratio of the intracell to intercell coupling strengths in the SSH chain. In the topologically nontrivial phase, the broad reflection structure around resonance is predominantly associated with an edge-state-related scattering channel, whereas the sharp reflection peaks in the non-resonant region originate from bulk-state modes. Increasing the number of atoms in the SSH chain leads to more bulk-state-related reflection peaks, providing a direct means to engineer the spectral structure. The accumulated phase between the two coupling points not only shifts the frequencies of completely reflected photons but also periodically modulates the entire reflection spectrum. Furthermore, we examine the influence of three typical types of disorder: hopping-strength disorder, atomic-frequency disorder, and accumulated-phase disorder. The broad reflection structure near resonance is found to be relatively robust against these disorders, while the narrow bulk-state-related reflection peaks are more sensitive to parameter fluctuations.

When atomic dissipation and asymmetric waveguide couplings are included, the system exhibits pronounced reflection nonreciprocity, even though the transmission remains reciprocal. At specific accumulated phases, perfect nonreciprocal reflection for resonant photons can be achieved by jointly tuning the SSH topological parameter and the atomic dissipation rate. This nonreciprocal response is also periodically controlled by the accumulated phase. These findings reveal the cooperative role of topological edge modes and giant-atom interference in single-photon transport, offering a theoretical route toward designing tunable nonreciprocal quantum devices and enabling directional photon manipulation in quantum networks.