We devise and analyze a Hybrid high-order (HHO) method to discretize unilateral and bilateral contact problems with Tresca friction in small strain elasticity. The nonlinear frictional contact conditions are enforced weakly by means of a consistent Nitsche's technique with symmetric, incomplete, and skew-symmetric variants. The present HHO-Nitsche method supports polyhedral meshes and delivers optimal energy-error estimates for smooth solutions under some minimal thresholds on the penalty parameters for all the symmetry variants. An explicit tracking of the dependency of the penalty parameters on the material coefficients is carried out to identify the robustness of the method in the incompressible limit, showing the more advantageous properties of the skew-symmetric variant. 2D and 3D numerical results including comparisons to benchmarks from the literature and to solutions obtained with an industrial software, as well as a prototype for an industrial application, illustrate the theoretical results and reveal that in practice the method behaves in a robust manner for all the symmetry variants in Nitsche's formulation.