Nonlinear manifold hyper-reduced order models provide an efficient framework for approximating nonlinear finite element problems by expressing the solution in a low-dimensional latent space. In nonlinear computational homogenization, this structure arises naturally because the microscopic response is driven by a low-dimensional macroscopic strain. The efficiency of these models relies on hyper-reduction techniques based on weighted cubature rules, but standard approaches use weights that are fixed a priori and remain constant over the entire solution manifold, even though the integrand itself evolves on a nonlinear low-dimensional manifold. This work introduces the Manifold-Adaptive Weight Empirical Cubature Method (MAW-ECM), in which cubature weights are allowed to vary smoothly along the nonlinear solution manifold. The method is built on subspace-adaptive hyper-reduction strategies but removes their main limitation: the lack of continuity of the weights when transitioning between local reduced spaces. In nonlinear homogenization and multiscale problems, such discontinuities can lead to a loss of robustness and convergence of the reduced-order solver. MAW-ECM constructs a smooth family of sparse cubature rules by progressively eliminating weights from an initial empirical cubature set and redistributing the remaining weights to satisfy integration constraints. Smoothness across the manifold is enforced through regularization terms based on the graph Laplacian of sampled manifold states, which couples neighboring reduced configurations and prevents abrupt weight variations. Numerical results in nonlinear computational homogenization show that the nonlinear manifold reduction can be driven by a number of latent variables comparable to the number of macroscopic strain components in large-strain hyperelasticity, and only slightly larger in path-dependent materials. Consequently, the number of selected integration points scales with this intrinsic dimensionality, yielding reductions of up to three orders of magnitude compared to conventional hyper-reduced order models.