We study the impact of accretion on the equation of state and on the global properties of neutron stars in the framework of the nuclear energy-density functional theory. Considering ashes made of ^56^Fe, we calculate the equations of state using the Brussels-Montreal generalised Skyrme functionals BSk19, BSk20, and BSk21, that have been already employed for determining the crustal heating for the same ashes. Unified equations of state for catalyzed neutron stars based on the same functionals are also available. All regions of accreting neutron stars are treated in a unified and thermodynamically consistent way. The equation of state of accreted neutron-star crusts is found to be significantly stiffer than that of catalyzed matter and, for this reason, accreting neutron stars have larger radii. However, their crustal moment of inertia and their tidal deformability are hardly changed provided density discontinuities at the interface between adjacent crustal layers are properly taken into account. The enhancement of the stiffness of the equation of state of accreting neutron stars is mainly a consequence of nuclear shell effects, thus confirming the importance of a quantum treatment. With our previous calculations of crustal heating using the same functionals, we have thus obtained consistent microscopic inputs for simulations of accreting neutron stars.