In this paper (arXiv:1403.4773) we construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel’d double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space- and time-like kappa-AdS and dS quantum algebras. The resulting non-commutative spacetime is a nonlinear cosmological constant deformation of the kappa-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel’d–Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.