We study the Lyapunov spectrum scaling of thermal weakly-nonlinear unitary maps in the presence of quasiperiodic potentials. We search for the crossover from long-range to short-range scaling as the localization length ξ decreases and compare the details to the case of uncorrelated Anderson disorder [Phys. Rev. Res. 6 L012064 (2024)]. A comparative statistical analysis of the eigenstates for the linear case shows that quasiperiodicity has a stronger localization impact at the same value of ξ. Therefore we expect that the scaling crossover should be enhanced as well. However, the numerical analysis shows that it is strongly delayed as compared to Anderson disorder, and is observed at anomalously small values of ξ. These findings hint at the potential impact of long range correlations of quasiperiodic localized eigenstates, which persist in the presence of interactions even in the case of integrability breaking and thermalization.