The UltraViolet EXplorer (UVEX) is a wide-field ultraviolet space telescope selected as a NASA Medium-Class Explorer (MIDEX) mission for launch in 2030. UVEX will undertake deep, cadenced surveys of the entire sky to probe low mass galaxies and explore the ultraviolet (UV) time-domain sky, and it will carry the first rapidly-deployable UV spectroscopic capability for a broad range of science applications. One of UVEX’s prime objectives is to follow up gravitational wave (GW) binary neutron star mergers as targets of opportunity (ToOs), rapidly scanning across their localization regions to search for their kilonova (KN) counterparts. Early-time multiband ultraviolet light curves of KNe are key to explaining the interplay between jet and ejecta in binary neutron star mergers. Owing to high Galactic extinction in the ultraviolet and the variation of GW distance estimates over the sky, the sensitivity to kilonovae can vary significantly across the GW localization and even across the footprint of a single image given UVEX’s large fields of view. Good ToO observing strategies to trade off between area and depth are neither simple nor obvious. We present an optimal strategy for GW follow-up with UVEX in which exposure time is adjusted dynamically for each field individually to maximize the overall probability of detection. We model the scheduling problem using the expressive and powerful mathematical framework of mixed integer linear programming (MILP), and employ a state-of-the-art MILP solver to automatically generate observing plan timelines that achieve high probabilities of kilonova detection. We have implemented this strategy in an open source astronomical scheduling software package called the Multi-Mission Multi-Messenger Observation Planning Toolkit (M4OPT), on GitHub at https://github.com/m4opt/m4opt. This is the data release accompanying the paper.