We present the radial profiles of the Hβ, Mg and Fe line-strength indices for a sample of eight spiral galaxies with a low-surface-brightness stellar disc and a bulge. The correlations between the central values of the line-strength indices and velocity dispersion are consistent with those known for early-type galaxies and bulges of high-surface-brightness galaxies. The age, metallicity and α/Fe enhancement of the stellar populations in the bulge-dominated region are obtained using stellar population models with variable element abundance ratios. Almost all the sample bulges are characterized by a young stellar population, ongoing star formation and a solar α/Fe enhancement. Their metallicity spans from high to subsolar values. No significant gradient in age and α/Fe enhancement is measured, whereas a negative metallicity gradient is found only in a few cases. These properties suggest that a pure dissipative collapse cannot explain the formation of all the sample bulges and that other phenomena, such as mergers or acquisition events, need to be invoked. Such a picture is also supported by the lack of a correlation between the central value and the gradient of the metallicity in bulges with very low metallicity. The stellar populations of the bulges hosted by low-surface-brightness discs share many properties with those of high-surface-brightness galaxies. Therefore, they are likely to have common formation scenarios and evolution histories. A strong interplay between bulges and discs is ruled out by the fact that, in spite of being hosted by discs with extremely different properties, the bulges of low- and high-surface-brightness discs are remarkably similar.