  We consider a diffuse interface model which describes the motion of an
incompressible isothermal mixture of two immiscible fluids. This model consists
of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard
equation. Several results were already proven by two of the present authors.
However, in the two-dimensional case, the uniqueness of weak solutions was
still open. Here we establish such a result even in the case of degenerate
mobility and singular potential. Moreover, we show the strong-weak uniqueness
in the case of viscosity depending on the order parameter, provided that either
the mobility is constant and the potential is regular or the mobility is
degenerate and the potential is singular. In the case of constant viscosity, on
account of the uniqueness results we can deduce the connectedness of the global
attractor whose existence was obtained in a previous paper. The uniqueness
technique can be adapted to show the validity of a smoothing property for the
difference of two trajectories which is crucial to establish the existence of
an exponential attractor. The latter is established even in the case of
variable viscosity, constant mobility and regular potential.
