Biomimetic natural biomaterial (BNBM) nanocomposite scaffolds for bone replacement can reduce the rate of implant failure and the associated risks of post-surgical complications for patients. Traditional bone implants, like allografts, and autografts, have limitations, such as donor site morbidity and potential patient inflammation. Over two million bone transplant procedures are performed yearly, and success varies depending on the material used. This emphasizes the importance of developing new biomaterials for bone replacement. Innovative BNBM nanocomposites for modern bone fabrication can promote the colonization of the desired cellular components and provide the necessary mechanical properties. Recent studies have highlighted the advantages of BNBM nanocomposites for bone replacement; therefore, this review focuses on the application of cellulose, chitosan, alginates, collagen, hyaluronic acid, and synthetic polymers enhanced with nanoparticles for the fabrication of nanocomposite scaffolds used in bone regeneration and replacement. This work outlines the most up-to-date overview and perspectives of selected promising BNBM nanocomposites for bone replacement that could be used for scaffold fabrication and replace other biomorphic materials such as metallics, ceramics, and synthetic polymers in the future. In summary, the concluding remarks highlight the advantages and disadvantages of BNBM nanocomposites, prospects, and future directions for bone tissue regeneration and replacement.