A graphene sheet biased with a drift electric current offers a unique opportunity to attain unidirectional, backscattering-immune, and subwavelength light propagation, as proposed by T. A. Morgado and M. G. Silveirinha [ACS Photonics 5, 4253 (2018)]. Here, we investigate in detail the impact of the intrinsic nonlocal response of graphene in the dispersion characteristics of the current-driven plasmons supported by single-layer and double-layer graphene systems. It is theoretically shown that even though the nonlocal effects weaken the spectral asymmetry of the plasmon dispersion, the studied platforms can support unidirectional backscattering-immune guided modes. Our analysis also confirms that the drift-current bias can effectively pump the graphene plasmons and enhance the propagation distance. Moreover, it is shown that the nonreciprocity and optical isolation can be boosted by pairing two drift-current-biased graphene sheets due to the enhanced radiation drag by the drifting electrons.