In the face of the problem of waste disposal in the demolition of concrete structures, a composite wall composed of recycled concrete bricks and fly ash blocks was proposed, and based on the previous thermal performance research, its axial compression performance were further studied. Four types of walls were designed and constructed: (1) clay brick masonry (CBM), (2) recycled concrete brick masonry (RBM), (3) bilateral clay bricks masonry with coal-ash blocks sandwich insulation wall (CFCM), and (4) bilateral recycled concrete bricks masonry with coal-ash blocks sandwich insulation wall (RFRM). The test results showed that recycled concrete brick masonry exhibited a higher bearing capacity than clay brick masonry. The ultimate load of RBM was 15% higher than that of CBM. Moreover, the ultimate load of CFCM was 21% higher than that of CBM. Following the addition of sandwich coal-ash blocks in RBM, its ultimate load increased by over 42% than that of CBM. Following the addition of coal-ash blocks sandwich in both clay and recycled concrete bricks masonry, both the bearing capacity and strain exhibited improvement, the yielding load and compressive strength of them increased. Thus, it could be concluded that coal-ash blocks improved its bearing capacity. Based on the analysis of the axial compression tests, a theoretical computational model was developed and a computational expression to explain the compressive bearing capacity of a two-sided brick with coal-ash blocks sandwich insulation wall. Comparisons between the test ultimate loads (FT) and the estimated ultimate loads (FE) confirmed the accuracy of the theoretical calculation model for the compressive bearing capacity. Thus, theoretical computational models are highly recommended for the design of two-sided bricks with insulating walls constructed from coal-ash blocks being sandwiched together. This study provides a theoretical basis for the engineering application of recycled concrete brick wall and fly ash block composite wall.