We present near-infrared observations of 42 gravitationally lensed galaxies obtained in the framework of the KMOS Lensed Emission Lines and VElocity Review (KLEVER) Survey, a programme aimed at investigating the spatially resolved properties of the ionized gas in 1.2<z<2.5 galaxies by means of a full coverage of the YJ, H, and K near-infrared bands. Detailed metallicity maps and gradients are derived for a subsample of 28 galaxies from reconstructed source-plane emission-line maps, exploiting the variety of different emission-line diagnostics provided by the broad wavelength coverage of the survey. About 85 per cent of these galaxies are characterized by metallicity gradients shallower than 0.05dex/kpc and 89 per cent are consistent with a flat slope within 3{sigma} (67 per cent within 1{sigma}), suggesting a mild evolution with cosmic time. In the context of cosmological simulations and chemical evolution models, the presence of efficient feedback mechanisms and/or extended star formation profiles on top of the classical 'inside-out' scenario of mass assembly is generally required to reproduce the observed flatness of the metallicity gradients beyond z~1. Three galaxies with significantly (>3{sigma}) 'inverted' gradients are also found, showing an anticorrelation between metallicity and star formation rate density on local scales, possibly suggesting recent episodes of pristine gas accretion or strong radial flows in place. Nevertheless, the individual metallicity maps are characterized by a variety of different morphologies, with flat radial gradients sometimes hiding non-axisymmetric variations on kpc scales, which are washed out by azimuthal averages, especially in interacting systems or in those undergoing local episodes of recent star formation.