This is the second paper of a series aimed to study the stellar kinematics and population properties of bulges in highly inclined barred galaxies. In this work, we carry out a detailed analysis of the stellar age, metallicity and [Mg/Fe] of 28 highly inclined (i > 65°) disc galaxies, from S0 to S(B)c, observed with the SAURON integral-field spectrograph. The sample is divided into two clean samples of barred and unbarred galaxies, on the basis of the correlation between the stellar velocity and h3 profiles, as well as the level of cylindrical rotation within the bulge region. We find that while the mean stellar age, metallicity and [Mg/Fe] in the bulges of barred and unbarred galaxies are not statistically distinct, the [Mg/Fe] gradients along the minor axis (away from the disc) of barred galaxies are significantly different than those without bars. For barred galaxies, stars that are vertically further away from the mid-plane are in general more [Mg/Fe]-enhanced and thus the vertical gradients in [Mg/Fe] for barred galaxies are mostly positive, while for unbarred bulges the [Mg/Fe] profiles are typically negative or flat. This result, together with the old populations observed in the barred sample, indicates that bars are long-lasting structures, and therefore are not easily destroyed. The marked [Mg/Fe] differences with the bulges of unbarred galaxies indicate that different formation/evolution scenarios are required to explain their build-up, and emphasizes the role of bars in redistributing stellar material in the bulge-dominated regions.