To identify the origin of the XY spin dimensionality in the bilayered system {Cu₄(tetrenH₅)[W(CN)₈] ₄·7.2H₂O}_n (WCuT) we use a combination of single-crystal experiments (bulk magnetization, neutron flipping ratio, electron magnetic resonance, neutron diffraction) and theoretical modeling (exchange-charge model of the crystal field, dipolar energy, and density functional calculations). Our experiments show that the magnetic properties of WCuT are anisotropic and two-dimensional correlations build up below 70 K. The hard anisotropy axis is perpendicular to the layers (b axis) and a small anisotropy within the ac layers is present. Modeling of the crystal field validates treatment of tungsten and copper as spin S=12 ions with anisotropic g values. The local magnetic anisotropy results from the common action of the crystal field and spin-orbit coupling and is along the c axis for both ions. Density functional calculations identify the origin of the ferromagnetic exchange in different energies and symmetries of the tungsten- and copper-dominated orbitals and anticipate different exchange couplings across the apical (along the b axis) and equatorial (in the ac plane) Cu-CN-W bridges due to difference in the hybridization efficiency. Calculation of the dipolar energy for various spin configurations suggests that dipolar interactions play a decisive role in the ac-planar anisotropy in this system. We propose that the effective XY spin dimensionality in WCuT is established by a combination of the axial local anisotropy of the W and Cu ions and the long-range magnetic dipolar interactions on the bilayered square lattice.