The present study investigates dumbbell-shaped nematic liquid crystal shells. Using molecular dynamics (MD) simulations, we consider the effects of an external electric field on nematic ordering by computing the average molecular alignment's time evolution and equilibrium configuration. We show that the number and location of topological defects are strongly affected by the external field, with the orientational ordering's equilibrium configuration depending on field direction about the shell's long axis. For a transverse external field, it is verified that the defect rearrangement presents a non-linear dynamics, with a field independent characteristic time scale delimiting the short and long time regimes. Effects associated with varying the shell's Gaussian curvature are also analyzed.