We investigate torsional chiral magnetic effect (TCME) induced by skyrmion-vortex textures in the A phase of the superfluid $^3$He. In $^3$He-A, Bogoliubov quasiparticles around point nodes behave as Weyl fermions, and the nodal direction represented by the $ell$-vector may form a spatially modulated texture. $ell$-textures generate a chiral gauge field and a torsion field directly acting on the chirality of Weyl-Bogoliubov quasiparticles. It has been clarified by G. E. Volovik [Pisma Zh. Eksp. Teor. Fiz. {bf 43}, 428 (1986)] that, if the $ell$-vector is twisted, the chiral gauge field is responsible for the chiral anomaly, leading to an anomalous current along ${ell}$. Here we show that, even for non-twisted $ell$-vector fields, a torsion arising from $ell$-textures brings about contributions to the equilibrium currents of Weyl-Bogoliubov quasiparticles along ${rm curl}{ell}$. This implies that while the anomalous current appears only for the twisted (Bloch-type) skyrmion of the $ell$-vector, the extra mass current due to TCME always exists regardless of the skyrmion type. Solving the Bogoliubov-de Gennes equation, we demonstrate that both Bloch-type and N{e}el-type skyrmions induce chiral fermion states with spectral asymmetry, and possess spatially inhomogeneous structures of Weyl bands in the real coordinate space. Furthermore, we discuss the contributions of Weyl-Bogoliubov quasiparticles and continuum states to the mass current density in the vicinity of the topological phase transition. In the weak coupling limit, continuum states give rise to backflow to the mass current generated by Weyl-Bogoliubov quasiparticles, which makes a non-negligible contribution to the orbital angular momentum. As the topological transition is approached, the mass current density is governed by the contribution of continuum states.
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