We study linear continuous-time systems with fast-varying almost periodic coefficients that are piecewise-continuous in time. Recently, a constructive time-delay approach to periodic averaging of systems with a single fast timescale was introduced and employed to averaging of systems with small time-varying delays (of the order of the small parameter). In this paper we present a novel transformation of the fast varying coefficients. This transformation is suitable for averaging over multiple timescales , and is applicable to averaging of systems with constant delays, where the value of delay is not small (i.e. essentially larger than the small parameter). We carry out stability analysis by employing time-varying Lyapunov functions (or functionals for the delayed case). The analysis leads to LMI conditions that are always feasible for small enough parameters. Numerical examples demonstrate the efficiency of the proposed approach and its conservatism.