The nonlinear evolution of magnetic reconnection in collisionless and weakly collisional regimes is analyzed on the basis of a two-dimensional incompressible fluid model. The initial equilibria are unstable to tearing modes. In the limit where the stability parameter Δ' is relatively large, the mode structure is characterized by global convective cells. It is found that the system exhibits a quasi-explosive time behavior in the early nonlinear stage where the fluid displacement is larger than the inertial skin depth but smaller than the typical size of the convective cells. The reconnection time is an order of magnitude shorter than the Sweet-Parker time for values of the inertial skin depth, of the ion Larmor radius and of the magnetic Reynolds number typical of the core of magnetic fusion experiments. The reconnection process is accompanied by the formation of a current density sub-layer narrower than the skin depth. In the strict dissipationless limit, this sublayer shrinks indefinitely in time. Physical mechanisms limiting this tendency to a singular current density profile are also discussed.