This paper presents a scaled reformulation of a robust second-order Discontinuous Galerkin (DG2) solver for the Shallow Water Equations (SWE), with guiding principles on how it can be naturally extended to fit into the multiresolution analysis of multiwavelets (MW). Multiresolution analysis applied to the flow and topography data enables the creation of an adaptive MWDG2 solution on a non-uniform grid. The multiresolution analysis also permits control of the adaptive model error by a single user-prescribed parameter. This results in an adaptive MWDG2 solver that can fully exploit the local (de)compression of piecewise-linear modelled data, and from which a first-order finite volume version (FV1) is directly obtainable based on the Haar wavelet (HFV1) for local (de)compression of piecewise-constant modelled data. The behaviour of the adaptive HFV1 and MWDG2 solvers is systematically studied on a number of well-known hydraulic tests that cover all elementary aspects relevant to accurate, efficient and robust modelling. The adaptive solvers are run starting from a baseline mesh with a single element, and their accuracy and efficiency are measured referring to standard FV1 and DG2 simulations on the uniform grid involving the finest resolution accessible by the adaptive solvers. Our findings reveal that the MWDG2 solver can achieve the same accuracy as the DG2 solver but with a greater efficiency than the FV1 solver due to the smoothness of its piecewise-linear basis, which enables more aggressive coarsening than with the piecewise-constant basis in the HFV1 solver. This suggests a great potential for the MWDG2 solver to efficiently handle the depth and breadth in resolution variability, while also being a multiresolution mesh generator. Accompanying model software and simulation data are openly available online.