Nonreciprocal propagation of spin waves provides the possibility to design novel functional magnonic devices. In the present paper, based on micromagnetic simulation, we investigate the properties of the spin waves propagating within magnetic domain walls in a synthetic antiferromagnetic bilayer structure and demonstrate a giant frequency nonreciprocity, of which not only the magnitude but also the sign can be efficiently controlled by an external magnetic field. This results from the field-induced modification of the hybridization between the domain-wall spin wave channels from different magnetic layers. The unidirectional excitation of the gigahertz hybridized domain-wall spin waves by a nanostripline antenna is also demonstrated. Our finding suggests the nanoscale antiferromagnetic domain walls should be a promising platform for programmable nonreciprocal spin-wave devices.