We investigate theoretically the dynamics of two quasidegenerate mechanical modes coupled through an open quantum two-level system. A mean-field approach shows that by engineering the retarded response of the two-level system with a coherent drive, the non-Hermitian mechanical spectrum exhibits an exceptional degeneracy point where the two modes coalesce. We show that this degeneracy can be exploited to manipulate the vectorial polarization of the mechanical oscillations. We find that adiabatically varying the detuning and the intensity of the drive induces a rotation of the mechanical polarization, which enables the topological and chiral actuation of one mode from the other. This topological manifestation of the degeneracy is further supported by quantum-jump Monte Carlo simulations to account for the strong quantum fluctuations due to the spontaneous emission of the two-level system. Our presentation focuses on a promising realization based on flexural modes of a carbon-nanotube cantilever coupled to a single-molecule electric dipole irradiated by a laser