Preexisting nuclei are often invoked in situations where the observed kinetics of a phase transition is much faster than one would expect from the classical theory of homogeneous nucleation. We examine analytically the time-dependent effects produced by such nuclei. Corrections to the number of nucleated particles, the time lag, and the transient rate of homogeneous nucleation are derived. For a high nucleation barrier the results are expected to be asymptotically accurate, both for the continuous ("Zeldovich-Frenkel") and the discrete ("Becker-Doring") nucleation models. This is confirmed by comparison with exact expressions available for a parabolic barrier in the continuous case, and with numerical solutions of the discrete Master equation. The probability formulation of the nucleation problem is also considered and the distribution of waiting times to defect the first "successful" nucleus is obtained.