A mathematical model of dynamic processes in a composite plate of prestressed piezoelectric (PE) and piezomagnetic (PM) layers is proposed. It is assumed that the initial strain state in the components of the plate is uniform and is induced by the action of initial mechanical stresses. In the quasi-static approximation, problems of shear horizontally polarized surface acoustic waves (SH-SAW) propagation in a PE/PM plate made of prestressed materials are considered. It is assumed that the materials of the plate layers in their natural state (NS) have symmetry class 6mm. The adhesion conditions are satisfied at the interface between the layers. The external surfaces of the heterostructure are in contact with the vacuum and are free from mechanical stress. Depending on the nature of the electrical and magnetic conditions specified on the external surfaces, four types of problems are considered. Using the example of a problem with electrically closed and magnetically open conditions on the external surfaces of a PZT-5H/CoFe₂O₄ plate, the features of the influence of the nature and magnitude of the initial mechanical effects on the velocities of SH waves are investigated. The possibility of changing the velocity characteristics of SAWs due to the initial mechanical effects is shown. The results are presented in dimensionless parameters and may be of significant interest for optimizing the structure of new materials in the development and design of devices and equipment operating on SH-SAW.