[Abridged] Using VLT/FORS2 spectroscopy, we have studied the properties of the central stellar populations of a sample of 38 nucleated early-type dwarf (dE) galaxies in the Virgo Cluster. We find that these galaxies do not exhibit the same average stellar population characteristics for different morphological subclasses. The nucleated galaxies without discs are older and more metal poor than the dEs with discs . The alpha-element abundance ratio appears consistent with the solar value for both morphological types. Besides a well-defined relation of metallicity and luminosity, we also find a clear anti-correlation between age and luminosity. More specifically, there appears to be a bimodality: brighter galaxies, including the discy ones, exhibit significantly younger ages than fainter dEs. Therefore, it appears less likely that fainter and brighter dEs have experienced the same evolutionary history, as the well-established trend of decreasing average stellar age when going from the most luminous ellipticals towards low-luminosity Es and bright dEs is broken here. The older and more metal-poor dEs could have had an early termination of star formation activity, possibly being primordial galaxies in the sense that they have formed along with the protocluster or experienced very early infall. By contrast, the younger and relatively metal-rich brighter dEs, most of which have discs, might have undergone structural transformation of infalling disc galaxies.
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