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We study the off-equilibrium critical phenomena across a hysteretic first-order transition in disordered athermal systems. The study focuses on the zero temperature random field Ising model (ZTRFIM) above the critical disorder for spatial dimensions $d=2,3,$ and $4$. We use Monte Carlo simulations to show that disorder suppresses critical slowing down in phase ordering time for finite-dimensional systems. The dynamic hysteresis scaling, the measure of explicit finite-time scaling, is used to subsequently quantify the critical slowing down. The scaling exponents in all dimensions increase with disorder strength and finally reach a stable value where the transformation is no longer critical. The associated critical behavior in the mean-field limit is very different, where the exponent values for various disorders in all dimensions are similar. The non-mean-field exponents asymptotically approach the mean-field value ($Υ\approx 2/3$) with increase in dimensions. The results suggest that the critical features in the hysteretic metastable phase are controlled by inherent mean-field spinodal instability that gets blurred by disorder in low-dimension athermal systems.