  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
$\Lambda$ 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; their flat limit $\Lambda\to 0$ leads to
a twisted quantum Poincar\'e algebra. The resulting non-commutative spacetime
is a nonlinear $\Lambda$-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.
