NGC 2419 is a peculiar Galactic globular cluster offset from the others in the size-luminosity diagram, and showing several chemical abundance anomalies. Here, we present Strömgren uvby photometry of the cluster. Using the gravity- and metallicity-sensitive c1 and m1 indices, we identify a sample of likely cluster members extending well beyond the formal tidal radius. The estimated contamination by cluster non-members is only one per cent, making our catalogue ideally suited for spectroscopic follow-up. We derive photometric [Fe/H] of red giants, and depending on which metallicity calibration from the literature we use, we find reasonable to excellent agreement with spectroscopic [Fe/H], both for the cluster mean metallicity and for individual stars. We demonstrate explicitly that the photometric uncertainties are not Gaussian and this must be accounted for in any analysis of the metallicity distribution function. Using a realistic, non-Gaussian model for the photometric uncertainties, we find a formal internal [Fe/H] spread of σ=0.11+0.02-0.01 dex. This is an upper limit to the cluster’s true [Fe/H] spread and may partially, and possibly entirely, reflect the limited precision of the photometric metallicity estimation and systematic effects. The lack of correlation between spectroscopic and photometric [Fe/H] of individual stars is further evidence against a [Fe/H] spread on the 0.1 dex level. Finally, the CN-sensitive δ4, among other colour indices, anti-correlates strongly with magnesium abundance, indicating that the second-generation stars are nitrogen enriched. The absence of similar correlations in some other CN-sensitive indices supports the second generation being enriched in He, which in these indices approximately compensates the shift due to CN. Compared to a single continuous distribution with finite dispersion, the observed δ4 distribution of red giants is slightly better fit by two distinct populations with no internal spread, with the nitrogen-enhanced second generation accounting for 53 ± 5 per cent of stars. Despite its known peculiarities, NGC 2419 appears to be very similar to other metal-poor Galactic globular clusters with a similarly nitrogen-enhanced second generation and little or no variation in [Fe/H], which sets it apart from other suspected accreted nuclei such as ωCen.