We theoretically investigate topological spin transport of the magnon polarons in a bilayer magnet with two-dimensional square lattices. Our theory is motivated by recent reports on the van der Waals magnets which show the reversible electrical switching of the interlayer magnetic order between antiferromagnetic and ferromagnetic orders. The magnetoelastic interaction opens band gaps and allows the interband transition between different excitation states. In the layered antiferromagnet, due to the interband transition between the magnon-polaron states, the spin Berry curvature which allows the topological spin transport occurs even if the time-reversal symmetry is preserved. We find that the spin Berry curvature in the layered antiferromagnet is very large due to the small energy spacing between two magnonlike states. As a result, the spin Nernst conductivity shows a sudden increase (or decrease) at the phase transition point between ferromagnetic and antiferromagnetic phases. Our results suggest that the ubiquity of tunable topological spin transports in two-dimensional magnetic systems.