The existence of star-to-star light-element abundance variations (multiple populations, MPs) in massive Galactic and extragalactic star clusters older than about 2 Gyr is by now well established. Photometry of red giant, branch (RGB) stars has been and still is instrumental in enabling the detection and characterization of cluster MPs, through the appropriate choices of filters, colours, and colour combinations that are mainly sensitive to N and to a lesser degree C stellar surface abundances. An important issue not yet properly addressed is that the translation of the observed widths of the cluster RGBs to abundance spreads must account for the effect of the first dredge-up on the surface chemical patterns, hence on the spectral energy distributions of stars belonging to the various MPs, We have filled this gap by studying theoretically the impact of the dredge-up on the predicted widths of RGBs in clusters hosting MPs. We find that for a given initial range of N abundances, the first dredge-up reduces the predicted RGB widths in N-sensitive filters compared to the case when its effect on the stellar spectral energy distributions is not accounted for. This reduction is a strong function of age and has also a dependence on metallicity. The net effect is an underestimate of the initial N-abundance ranges from RGB photometry if the first dredge-up is not accounted for in the modelling, and also the potential determination of spurious trends of N-abundance spreads with age.