The time evolution in quantum many-body systems after external excitations is attracting high interest in many fields. The theoretical modeling of these processes is challenging, and the only rigorous quantum-dynamics approach that can treat correlated fermions in two and three dimensions is nonequilibrium Green functions (NEGF). However, NEGF simulations are computationally expensive due to their $T^3$-scaling with the simulation duration $T$. Recently, $T^2$-scaling was achieved with the generalized Kadanoff--Baym ansatz (GKBA), for the second-order Born (SOA) selfenergy, which has substantially extended the scope of NEGF simulations. In a recent Letter [Schlunzen textit{et al.}, Phys. Rev. Lett. textbf{124}, 076601 (2020)] we demonstrated that GKBA-NEGF simulations can be efficiently mapped onto coupled time-local equations for the single-particle and two-particle Green functions on the time diagonal, hence the method has been called G1--G2 scheme. This allows one to perform the same simulations with order $T^1$-scaling, both for SOA and $GW$ selfenergies giving rise to a dramatic speedup. Here we present more details on the G1--G2 scheme, including derivations of the basic equations including results for a general basis, for Hubbard systems and for jellium. Also, we demonstrate how to incorporate initial correlations into the G1--G2 scheme. Further, the derivations are extended to a broader class of selfenergies, including the $T$ matrix in the particle--particle and particle--hole channels, and the dynamically screened-ladder approximation. Finally, we demonstrate that, for all selfenergies, the CPU time scaling of the G1--G2 scheme with the basis dimension, $N_b$, can be improved compared to our first report: the overhead compared to the original GKBA, is not more than an additional factor $N_b$.
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