We investigate the flow of a viscous ferrofluid annulus surrounded by two nonmagnetic fluids in a Hele-Shaw cell when subjected to an external radial magnetic field. The interfacial pattern formation dynamics of the system is determined by the interplay of magnetic and surface tension forces acting on the inner and outer boundaries of the annulus, favoring the coupling of the disjoint interfaces. Mode-coupling analysis is employed to examine both linear and weakly nonlinear stages of the flow. Linear stability analysis indicates that the trailing and leading annular boundaries are coupled already at the linear regime, revealing that perturbations arising in the outer interface may induce the emergence of deformed structures in the inner boundary. Moreover, second-order weakly nonlinear analysis is utilized to identify key nonlinear morphological features of the ferrofluid annulus. Our theoretical results show that linear, n-fold symmetric annular patterns having rounded edges are replaced by nonlinear polygonal-like shapes, presenting fairly sharp fingers. It is found that, as opposed to the linear patterns, the nonlinear peaky structures reach a stationary state, characterized by a growth saturation process induced by nonlinear effects. Furthermore, the response of the ferrofluid ring to changes in the thickness of the annulus, in the relative strength of magnetic and surface tensions forces, as well as in the magnetic susceptibility of the ferrofluid material, are also discussed.