The non-equilibrium dynamics of isolated quantum statistical systems after a quench is characterised by the possible emergence of long-lived pre-thermal states, which were observed in experiments with ultra cold atomic gases in one spatial dimension. Depending on the parameters of the quench, dynamical transitions of collective nature, i.e., accompanied by dynamical scaling and aging, are expected to occur within these states, as demonstrated by a number of models in more than one spatial dimension. The universal features of the associated dynamic critical points turn out to be influenced by the gross features of the pre-quench state. As a result, the evolution of the order parameter and of the response and correlation functions can exhibit temporal crossovers between two universal dynamical scaling regimes governed by a quantum and a classical or by a Gaussian and a non-Gaussian prethermal fixed point, respectively, which we investigate analytically by using a combination of renormalization-group techniques and a non-perturbative limit. Together with a recent experiment, this demonstrate the rich variety of dynamical critical behaviours which emerge in the non-equilibrium dynamics of isolated quantum statistical systems.