Elastic-driven slender filaments subjected to compressive follower forces provide a synthetic way to mimic the oscillatory beating of biological flagella and cilia. Here, we use a continuum model to study the dynamical, nonlinear buckling instabilities that arise due to the action of nonconservative follower forces on a prestressed slender rod clamped at both ends and allowed to move in a fluid. Stable oscillatory responses are observed as a result of the interplay between the structural elastic instability of the inextensible slender rod, geometric constraints that control the onset of instability, energy pumped into the system by the active follower forces, and motion-driven fluid dissipation. Initial buckling instabilities are initiated by the effect of the follower forces and inertia; fluid drag subsequently allows for the active energy pumped into the system to be dissipated away and results in self-limiting amplitudes. By integrating the equations of equilibrium and compatibility conditions with linear constitutive laws, we compute the critical follower forces for the onset of oscillations, emergent frequencies of these solutions, and the postcritical nonlinear rod shapes for two forms of the drag force, namely linear Stokes drag and quadratic Morrison drag. For a rod with fixed inertia and drag parameters, the minimum (critical) force required to initiate stable oscillations depends on the initial slack and weakly on the nature of the drag force. Emergent frequencies and the amplitudes postonset are determined by the extent of prestress as well as the nature of the fluid drag. Far from onset, for large follower forces, the frequency of the oscillations can be predicted by evoking a power balance between the energy input by the active forces and the dissipation due to fluid drag.