Universal dynamics of a dilute Bose gas is studied within the kinetic regime where the time evolution of the mode occupation number is governed by a wave-Boltzmann equation. The universality manifests itself in the form of the dynamical evolution which can be a cascade in the context of Kolmogorov-Zakharov wave turbulence or a self-similar shift in time and space. Which is the case depends on the relevant global conservation laws and on the particular range of scales in which the respective dynamics takes place. A nonperturbative kinetic equation is derived by applying the Schwinger-Keldysh closed-time functional integral and the 2-particleirreducible formalism to an action of complex scalar Bose fields with quartic interaction. The resulting dynamic equation for two-point correlations are reduced to a wave-Boltzmann-type kinetic equation with an effective T-matrix which depends explicitly on the mode occupation numbers. This nonperturbative dependence on the solution occurs in the infrared regime where wave numbers are large and collective scattering of many particles prevails. Thus, the scaling analysis of wave turbulence theory can be applied. We explicitly calculate the effective T-matrix analytically taking into account an infrared cutoff for dealing with the infrared divergences. Our results show that the scaling behaviour of the T-matrix departs from the one predicted by naive dimensional counting due to the presence of the infrared cutoff. The Kolmogorov-Zakharov and the self-similar exponents are evaluated by power counting using our T-matrix and the results are confirmed by numerical integration of the wave-Boltzmann scattering integral. The scaling exponents governing the time evolution are determined by means of the global conservation laws and the kinetic equation. Depending on the scaling properties of the quasiparticle spectrum, the momentum cutoff scale in the infrared evolves critically slowed in time. The respective scaling exponent is universal in the nonperturbative regime regardless of whether the process is an inverse cascade or a self-similar shift towards the infrared. Thus, our results provide a general framework for classifying nonthermal fixed points in dilute ultracold Bose gases. They pave the way to a straightforward generalisation and application to trapped systems in different dimensionalities and to systems with more than one internal degree of freedom. Our results furthermore provide a possible interpretation of recent experimental results on wave turbulence in an ultracold Bose gas.