The transient dynamics of phase-locking in serially connected nanopillar spin-torque oscillators (STOs) is studied both analytically and numerically. A variety of transient behaviors are observed stemming from the high oscillator nonlinearity and the interplay between the damping to coupling strength ratio and the phase delay of the coupling. Non-Adlerian (ringing) dynamics is found to be the main regime of synchronization where the synchronization time depends strongly on the phase delay. Somewhat nonintuitively, sufficiently strong coupling can also destabilize the system, destroying the synchronized regime even for identical STOs. This transient behavior is also found to dominate when the STOs have different frequencies. These results highlight fundamental issues that must be considered in the design of serially synchronized STOs.