We investigate multicomponent superconductivity in twisted bilayer cuprates with order parameter $s+d_1{e}^{i{\varphi }_1}+d_2{e}^{i{\varphi }_2}$, where $s=s_1+s_2$ is the layer-symmetric $s$-wave pairing and $d_i$ is the $d$-wave pairing in layer $i$. When ${\varphi }_1-{\varphi }_2\ne 0,\pi$, this three-component state spontaneously breaks time-reversal and $C_4$ rotational symmetries and can realize a topologically nontrivial phase. Ginzburg-Landau analysis and self-consistent mean-field calculations show that this state is stabilized over a broad parameter regime. We further identify nematic Kerr anisotropy as a key signature distinguishing it from $s+id$ and $d_1+e^{i\varphi }{d}_2$, and find that its Josephson critical current remains sizable near the ${45}^{\circ }$ twist. Our results show that $s$-wave pairing does not preclude chiral topological superconductivity, pointing to twisted cuprates as a more robust platform than previously appreciated.