We present the Sherwood-Relics simulations, a new suite of large cosmological hydrodynamical simulations aimed at modelling the intergalactic medium (IGM) during and after the cosmic reionization of hydrogen. The suite consists of over 200 simulations that cover a wide range of astrophysical and cosmological parameters. It also includes simulations that use a new lightweight hybrid scheme for treating radiative transfer effects. This scheme follows the spatial variations in the ionizing radiation field, as well as the associated fluctuations in IGM temperature and pressure smoothing. It is computationally much cheaper than full radiation hydrodynamics simulations and circumvents the difficult task of calibrating a galaxy formation model to observational constraints on cosmic reionization. Using this hybrid technique, we study the spatial fluctuations in IGM properties that are seeded by patchy cosmic reionization. We investigate the relevant physical processes and assess their impact on the z > 4 Lyman-alpha forest. Our main findings are: (i) Consistent with previous studies patchy reionization causes large scale temperature fluctuations that persist well after the end of reionization, (ii) these increase the Lyman-alpha forest flux power spectrum on large scales, and (iii) result in a spatially varying pressure smoothing that correlates well with the local reionization redshift. (iv) Structures evaporated or puffed up by photoheating cause notable features in the Lyman-alpha forest, such as flat-bottom or double-dip absorption profiles.