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Deep Learning (DL) algorithms are becoming increasingly popular for the reconstruction of high-resolution turbulent flows (aka super-resolution). However, current DL approaches perform spatially uniform super-resolution - a key performance limiter for scalability of DL-based surrogates for Computational Fluid Dynamics (CFD). To address the above challenge, we introduce NUNet, a deep learning-based adaptive mesh refinement (AMR) framework for non-uniform super-resolution of turbulent flows. NUNet divides the input low-resolution flow field into patches, scores each patch, and predicts their target resolution. As a result, it outputs a spatially non-uniform flow field, adaptively refining regions of the fluid domain to achieve the target accuracy. We train NUNet with Reynolds-Averaged Navier-Stokes (RANS) solutions from three different canonical flows, namely turbulent channel flow, flat plate, and flow around ellipses. NUNet shows remarkable discerning properties, refining areas with complex flow features, such as near-wall domains and the wake region in flow around solid bodies, while leaving areas with smooth variations (such as the freestream) in the low-precision range. Hence, NUNet demonstrates an excellent qualitative and quantitative alignment with the traditional OpenFOAM AMR solver. Moreover, it reaches the same convergence guarantees as the AMR solver while accelerating it by 3.2-5.5x, including unseen-during-training geometries and boundary conditions, demonstrating its generalization capacities. Due to NUNet's ability to super-resolve only regions of interest, it predicts the same target 1024x1024 spatial resolution 7-28.5x faster than state-of-the-art DL methods and reduces the memory usage by 4.4-7.65x, showcasing improved scalability.