Time-dependent flow simulations around complex geometries pose a dual challenge: capturing transient flow physics while accurately representing intricate boundaries. This work addresses both aspects through an unfitted hybridizable discontinuous Galerkin (HDG) method that embeds exact NURBS geometries in fixed (typically Cartesian) meshes, extending high-order accuracy to transient incompressible Navier–Stokes equations without body-fitted mesh generation. The approach builds upon the HDG-NURBS enhanced finite element (NEFEM) framework for steady Stokes flows, employing NEFEM quadratures for exact integration in NURBS-intersected elements. The method preserves the HDG computational efficiency: global unknowns reside only on the mesh skeleton, with Dirichlet and Neumann conditions handled seamlessly at arbitrary polynomial order. Selective mesh refinement near boundaries combined with variable polynomial degrees efficiently resolves boundary layers, vortex structures, and other localised flow features in transient regimes. The method is validated on benchmark cases, including flow past cylinders and NACA aerofoils with varying mesh and polynomial degree configurations. Numerical results confirm accurate resolution of transient vortex shedding and boundary layer dynamics directly from CAD models, maintaining high-order accuracy despite arbitrary geometry-mesh intersections. The unfitted framework combines computational efficiency (avoiding remeshing and curved element generation) with the flexibility to handle complex geometries through simple Cartesian grids and strategic h/p refinement.