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RR Lyrae stars are useful chemical tracers thanks to the empirical relationship between their heavy-element abundance and the shape of their light curves. However, the consistent and accurate calibration of this relation across multiple photometric wavebands has been lacking. We have devised a new method for the metallicity estimation of fundamental-mode RR Lyrae stars in the Gaia optical $G$ and near-infrared VISTA $K_s$ wavebands by deep learning. First, an existing metallicity prediction method is applied to large photometric data sets, which are then used to train long short-term memory recurrent neural networks for the regression of the [Fe/H] to the light curves in other wavebands. This approach allows an unbiased transfer of our accurate, spectroscopically calibrated $I$-band formula to additional bands at the expense of minimal additional noise. We achieve a low mean absolute error of $0.1$ dex and high $R^2$ regression performance of $0.84$ and $0.93$ for the $K_s$ and $G$ bands, respectively, measured by cross-validation. The resulting predictive models are deployed on the Gaia DR2 and VVV inner-bulge RR Lyrae catalogs. We estimate mean metallicities of $-1.35$ dex for the inner bulge and $-1.7$ for the halo, which are significantly less than values obtained by earlier photometric prediction methods. Using our results, we establish a public catalog of photometric metallicities of over 60,000 Galactic RR Lyrae stars, and provide an all-sky map of the resulting RR Lyrae metallicity distribution. The software code used for training and deploying our recurrent neural networks is made publicly available in the open-source domain.