We revisit the decoupling of neutrinos in the early universe with flavour oscillations. We rederive the quantum kinetic equations which determine the neutrino evolution based on a BBGKY-like hierarchy, and include for the first time the full collision term, with both on- and off-diagonal terms for all relevant reactions. We focus on the case of zero chemical potential and solve these equations numerically. We also develop an approximate scheme based on the adiabatic evolution in the matter basis. In fact, the large difference between the oscillations and cosmological time scales allows to consider averaged flavour oscillations which can speed up the numerical integration by two orders of magnitude, when combined with a direct computation of the differential system Jacobian. The approximate numerical scheme is also useful to gain more insight into the physics of neutrino decoupling. Including the most recent results on plasma thermodynamics QED corrections, we update the effective number of neutrinos to $N_{mathrm{eff}} = 3.0440$. Finally we study the impact of flavour oscillations during neutrino decoupling on the subsequent primordial nucleosynthesis.
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