Hyperspectral imaging is one of the most promising techniques for intraoperative tissue characterisation. Snapshot mosaic cameras, which can capture hyperspectral data in a single exposure, have the potential to make a real-time hyperspectral imaging system for surgical decision-making possible. However, optimal exploitation of the captured data requires solving an ill-posed demosaicking problem and applying additional spectral corrections to recover spatial and spectral information of the image. In this work, we propose a deep learning-based image demosaicking algorithm for snapshot hyperspectral images using supervised learning methods. Due to the lack of publicly available medical images acquired with snapshot mosaic cameras, a synthetic image generation approach is proposed to simulate snapshot images from existing medical image datasets captured by high-resolution, but slow, hyperspectral imaging devices. Image reconstruction is achieved using convolutional neural networks for hyperspectral image super-resolution, followed by cross-talk and leakage correction using a sensor-specific calibration matrix. The resulting demosaicked images are evaluated both quantitatively and qualitatively, showing clear improvements in image quality compared to a baseline demosaicking method using linear interpolation. Moreover, the fast processing time of ~45ms of our algorithm to obtain super-resolved RGB or oxygenation saturation maps per image frame for a state-of-the-art snapshot mosaic camera demonstrates the potential for its seamless integration into real-time surgical hyperspectral imaging applications.