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Protecting and restoring natural spaces is critical in the face of climate risks and environmental change, whilst at the same time, access to natural space plays an important role in population health and well-being. Understanding visitation patterns to natural spaces aids planning, maintenance, and land use, and allows us to evaluate the impact of interventions designed to benefit both nature and society. While surveys can provide snapshots of information about visits to natural spaces, robustly measuring visitor patterns on broad scales remains a challenge. Moreover, we lack the tools required to provide visitation estimates under the range of scenarios involved in land use and natural space planning. In this research, we develop scalable tools to predict visitor counts along paths in the UK located in natural spaces using Machine Learning methods, expanding on previous work by the Office for National Statistics. We employ a range of linear, tree-based, and time series models trained on automated footplate counter data and test our models across a range of spatial and temporal scenarios. Our models demonstrate promising ability to replicate historical visitation patterns at many sites, suggesting data-driven methods could offer valuable insights into the sustainable management of natural spaces. We also highlight areas for future improvement, such as improving the spatial generalisability of the models, which could inform future visitation monitoring strategies. Finally, we use Explainable AI approaches to investigate the characteristics of natural space visitation, providing information for planning and interventions which we explore in this study using a storytelling approach.