It is known that in chiral magnets with intrinsic inversion symmetry breaking, two spin waves moving in opposite directions can propagate at different velocities, exhibiting a phenomenon called magnetochiral nonreciprocity, which allows for realizations of certain spin logic devices such as a spin-wave diode. Here, we theoretically demonstrate that the spin-wave nonreciprocity can occur without intrinsic bulk chirality in easy-cone ferromagnets and easy-cone antiferromagnets. Specifically, we show that nonlocal injection of a spin current from proximate normal metals to easy-cone magnets engenders a nonequilibrium chiral spin texture, on top of which spin waves exhibit nonreciprocity proportional to the injected spin current. In particular, the easy-cone ferromagnet is shown to support the spin-wave nonreciprocity without an external field, in contrast to the previously known easy-plane ferromagnetic counterpart that requires an external field, thereby providing a field-free means to manipulate the spin-wave nonreciprocity. One notable feature of the nonreciprocal spin waves is their gapless nature, which can lead to a large thermal rectification effect at sufficiently low temperatures.