The quasi-brittle nature of rocks challenges the basic assumptions of linear hydraulic fracture mechanics (LHFM): namely, linear elastic fracture mechanics and smooth parallel plates lubrication fluid flow inside the propagating fracture. We relax these hypotheses and investigate in details the growth of a plane-strain hydraulic fracture in an impermeable medium accounting for a rough cohesive zone and a fluid lag. In addition to a dimensionless toughness and the time-scale of coalescence of the fluid and fracture fronts governing the fracture evolution in the LHFM case, the solution now also depends on the ratio between the in-situ stress and material peak cohesive stress and the intensity of the flow deviation induced by aperture roughness (captured by a dimensionless power exponent). We show that the solution is appropriately described by a nucleation time-scale , which delineates the fracture growth into three distinct stages: a nucleation phase (), an intermediate stage () and late time () stage where convergence toward LHFM predictions finally occurs. A highly non-linear hydro-mechanical coupling takes place as the fluid front enters the rough cohesive zone which itself evolves during the nucleation and intermediate stages of growth. This coupling leads to significant additional viscous flow dissipation. As a result, the fracture evolution deviates from LHFM predictions with shorter fracture lengths, larger widths and net pressures. These deviations from LHFM ultimately decrease at late times () as the ratios of the lag and cohesive zone sizes with the fracture length both become smaller. The deviations increase with larger dimensionless toughness and larger ratio, as both have the effect of further localizing viscous dissipation near the fluid front located in the small rough cohesive zone. The convergence toward LHFM can occur at very late time compared to the nucleation time-scale (by a factor of hundred to thousand times) for realistic values of encountered at depth. The impact of a rough cohesive zone appears to be prominent for laboratory experiments and short in-situ injections in quasi-brittle rocks with ultimately a larger energy demand compared to LHFM predictions.