We reveal the generic characteristics of wave packet delocalization in two-dimensional nonlinear disordered lattices by performing extensive numerical simulations in two basic disordered models: the Klein-Gordon system and the discrete nonlinear Schr{o}dinger equation. We find that in both models (a) the wave packets second moment asymptotically evolves as $t^{a_m}$ with $a_m approx 1/5$ ($1/3$) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S.~Flach, Chem.~Phys.~{bf 375}, 548 (2010)], (b) chaos persists, but its strength decreases in time $t$ since the finite time maximum Lyapunov exponent $Lambda$ decays as $Lambda propto t^{alpha_{Lambda}}$, with $alpha_{Lambda} approx -0.37$ ($-0.46$) for the weak (strong) chaos case, and (c) the deviation vector distributions show the wandering of localized chaotic seeds in the lattices excited part, which induces the wave packets thermalization. We also propose a dimension-independent scaling between the wave packets spreading and chaoticity, which allows the prediction of the obtained $alpha_{Lambda}$ values.
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