Nonlinear projection-based reduced-order models (PROMs) deliver substantial computational speedups for large-scale problems in computational mechanics. However, using global affine trial spaces, accurate approximations for multiple regimes, moving features, or strong nonlinearities may require large reduced dimensions. This arises because the solution manifold exhibits a slowly decaying Kolmogorov n-width, making it difficult for a single low-dimensional linear subspace to capture the manifold accurately. Although augmenting the linear subspace with a nonlinear latent-space closure model improves approximation, accurately representing the manifold across multiple regimes may still require many retained modes. Building on local reduced-order bases, regression-based latent-space closure, and sampling-and-weighting strategies, this contribution investigates local nonlinear PROMs with regression-based latent-space closure for regime-dependent dynamics. Offline, snapshot data are partitioned into regions, and a local POD basis is constructed for each region. Within each region, the reduced-order basis is further decomposed into retained and discarded modes: the discarded modes do not directly contribute to the PROM's reduced dimension but preserve information associated with the local solution manifold. A local regression model is trained for each region to map retained coordinates to the corresponding discarded ones, with regression options including Gaussian-process regression, radial-basis-function interpolation, and artificial neural networks. Online, the governing equations are advanced via projection-based time integration using either Galerkin projection or LSPG residual minimization, while hyperreduction is performed through project-then-approximate sampling and weighting so that residual and Jacobian evaluations scale with a reduced mesh. For the CIMNE application, namely, nonlinear RVE homogenization, the proposed approximation approach is implemented using Galerkin projection with ECM hyperreduction. For the Stanford applications, which range from parametric Burgers dynamics to turbulent flow around the Ahmed body and parametric hypersonic flows, the local approximation method is implemented using LSPG projection combined with ECSW hyperreduction. By exploiting locality, these approaches enable faster and more accurate reduced-order models, requiring fewer retained modes and smaller hyper-reduced meshes than global methods.