The strong increase in computational power observed during the last few years has allowed to use Large Eddy Simulation (LES) for industrial configurations. Nevertheless, the time-to-solution is still too large for a daily use in the design phases. The objective of this work is to develop a new time integration method to reduce the time-to-solution of LES of incompressible flows by allowing the use of larger time step. The projection method, probably the most commonly used method in the context of LES of incompressible flow, is generally applied using explicit time advancement which constrains the time-step value for stability reasons (CFL and Fourier constraints). The time step can then be small with respect to the physical characteristic times of the studied flow. In this case, an implicit time advancement method, which is unconditionally stable, can be used. However, this leads to non-linear resolution of momentum equation which can strongly increase time-to-solution because of non-linear iterations inside a physical iteration. To relax the stability constraints while minimising the computational cost of an iteration, a linearised implicit time advancement based on Backward Differentiation Formula (BDF) scheme is proposed in this work. The linearisation is performed using an extrapolated velocity field based on the previous fields. This time integration is first evaluated on a turbulent pipe test case. It is observed a time-to-solution up to five times lower than the explicit time integration while keeping the same accuracy in terms of mean and fluctuating velocity fields. To incorporate this new time advancement method in the automatic mesh convergence developed in Part I, a time-step control method based on the local truncation error is used. The resulting automatic time-step and mesh procedure is evaluated on a turbulent round jet case and on PRECCINSTA configuration, a swirl burner which is a representative case of an industrial aeronautical injection system. This new procedure leads to a time-to-solution up to three times lower than the previous procedure, presented in Part I.