We investigate by means of two-dimensional incompressible magnetohydrodynamic (MHD) numerical simulations, the onset phase of the fast collisional magnetic reconnection regime that is supported by the formation of plasmoid chains when the Lundquist number S exceeds a critical value. The present study extends previous results obtained at magnetic Prandtl number Pm = 1 (Baty 2020) to a range of different Pm values. We use FINMHD code where a set of reduced visco-resistive MHD equations is employed to form two quasi-singular current layers as a consequence of the tilt instability. The results reinforce the conclusion that, a phase of sudden super-Alfvenic growth (when Pm is not too high) of plasmoid chains is obtained, following a previous quiescent phase during current sheet formation on a slower Alfvenic time scale. We compare our results with predictions from the general theory of the plasmoid instability. We also discuss the importance of this onset phase to reach the ensuing stochastic time-dependent reconnection regime, where a fast time-averaged rate independent of S is obtained. Finally, we briefly discuss the relevance of our results to explain the flaring activity in solar corona and internal disruptions in tokamaks.