Multispectral imaging systems are increasingly used in many scientific fields. However multispectral images generally present spectral and spatial limitations: the spectral information within each band is lacking because of spectral integration over the band, and the spatial resolution is limited due to the spatial convolution by spectrally variant Point Spread Functions which introduce a spatial variant blur. To address the ill-posed inverse problem of reconstruction from wideband images, we propose a new approach combining a precise instrument model for the degraded multispectral images together with a spectral approximation on a low-rank subspace of the object. The reconstruction is based on the minimization of a convex objective function composed of a data fidelity and an edge-preserving regularization term. The proposed half-quadratic algorithm alternates between the minimization of a quadratic and a separable problem, and we show that both closed-form solutions are available and tractable. Therefore, even with a non-stationary data model, the algorithm is very fast and results are obtained in a few seconds. Several tests are performed for multispectral data to be taken by MIRI, the mid-infrared imager of the future James Webb Space Telescope (JWST). The reconstruction results show a significant increase in spatial and spectral resolutions compared to state-of-the-art methods. Our proposed algorithm allows us to recover the spectroscopic information contained in the wideband multispectral images and to provide hyperspectral images with a homogenized spatial resolution over the entire spectral range.