High-order finite element discretizations can deliver superior accuracy and efficiency compared with low-order schemes, but their use is often limited by the difficulty of generating valid, well-shaped curved, body-fitted meshes for complex geometries. This work presents an unfitted, high-order, degree-adaptive Hybridizable Discontinuous Galerkin (HDG) method for incompressible flows that bypasses this meshing bottleneck. Boundaries and interfaces are represented exactly with NURBS that cut a typically Cartesian background mesh, enabling high-fidelity geometry treatment without constructing curved high-order meshes. Accurate integration over cut elements is performed using NURBS-enhanced finite element (NEFEM) techniques. Dirichlet and Neumann boundary conditions are imposed weakly and consistently at any polynomial order through standard HDG stabilization. The approach preserves the standard HDG architecture: only trace unknowns on the mesh skeleton are globally coupled, and no additional degrees of freedom are introduced on non-matching boundaries or interfaces. The resulting unfitted HDG–NEFEM framework (NURBS-CutHDG) combines simple grids, exact CAD geometry, and high-order accuracy, remaining reliable even with severely cut elements. Originally developed for two-fluid Stokes problems, it extends directly to incompressible Navier–Stokes flows while maintaining optimal convergence and robustness. Numerical results confirm the method’s accuracy and stability.