Your Department or research group may have a small amount of money available to support public engagement, but it is good practice to plan your engagement from the moment you apply for funding. Many funding agencies will allow you to add public engagement funds to your grant application.
University of Oxford
The University administers a Public Engagement Seed fund.
The Medical Research Council permits public engagement with research to be funded as part of its research grants. For those working in MRC funded units, institutes, and centres there is also the public engagement seed fund.
The learned societies sometimes have funding for area of interest. For example, The Royal Society of Chemistry, The British Society for Neuroendocrinology, The Society for Applied Microbiology, and The Physiological Society have schemes. The National Coordinating Centre for Public Engagement has funding links to more.
Medical Research Charities
Many medical issues have charities dedicated to research and awareness of the conditions. These charities sometimes support public engagement. For example, Alzheimers Research UK, and The Nation Centre for Replacement, Refinement and Reduction of Animals in Research.