In recent years, attention has turned to several numerical challenges that impact classical inf-sup stable mixed discretisations of the Navier-Stokes problem, particularly in convection-dominated flows. Numerical schemes for such problems may suffer from poor convergence rates and nonphysical oscillations in the discrete solution. A number of convective stabilisation techniques have been developed to address the issue of non-physical oscillations, but such stabilisation may weaken error estimates through the introduction of inverse powers of the viscosity. Schemes whose velocity error estimates are independent of such terms are called Reynolds-semi-robust [1, 2]. Many classical mixed methods also lack pressure-robustness [3]. Stemming from the observation that the velocity solution in the continuous Navier-Stokes problem is unaffected by the irrotational part of the body force, the notion of pressure-robustness is introduced in [4] to describe a scheme whose velocity error scales independently of the continuous pressure solution. We introduce a pressure-robust and Reynolds-semi-robust hybrid high-order (HHO) scheme for the incompressible unsteady Navier-Stokes problem on simplicial meshes, with a view of extending the scheme to general polytopic meshes. Pressure-robustness is ensured by a careful treatment of the velocity-pressure coupling, enforcing the divergence-free constraint pointwise. We prove well-posedness and convergence of the scheme, and present a velocity error estimate that is robust with respect to the pressure variable and optimal in both the convective and viscous limits. The optimal convergence and pressure-robustness of the scheme will be demonstrated through numerical examples.