A comprehensive study of the angular distributions in the bottom-baryon decays $\it{\Lambda}^\mathrm{0}_b\to\it{\Lambda}_c^+ h^-(h=\pi, K)$, followed by $\it{\Lambda}_c^+\to\it{\Lambda} h^+$ with $\it{\Lambda}\to \it{p} \pi^-$ or $\it{\Lambda}_c^+\to\it{p}\it{K}^0_\mathrm{S}$ decays, is performed using a data sample of proton-proton collisions corresponding to an integrated luminosity of $9~\mathrm{fb}^{-1}$ collected by the LHCb experiment at center-of-mass energies of 7, 8 and 13 $\mathrm{Te\kern -0.1em V}$. The decay parameters and the associated charge-parity ($C\!P$) asymmetries are measured, with no significant $C\!P$ violation observed. For the first time, the $\it{\Lambda}^\mathrm{0}_b \to \it{\Lambda}_c^+ h^-$ decay parameters are measured. The most precise measurements of the decay parameters $\alpha, \beta$ and $\gamma$ are obtained for $\it{\Lambda}_c^+$ decays and an independent measurement of the decay parameters for the strange-baryon $\it{\Lambda}$ decay is provided. The results deepen our understanding of weak decay dynamics in baryon decays.