The fact that they've phoned you, rather than sending someone to your house, would seem to suggest that they're just 'going through the motions'. (People with mental health problems are often known to the police. The police can't ignore complaints from them, as some might be genuine, but they know that their evidence isn't always to be relied upon).
Given it's simply a case of one person's word against another, it's extremely unlikely that the case would ever get anywhere near to a court. The only cost that might be involved would be the tenner that you could pay to the rail company to get hold of footage from their cameras, showing that you walked to your car without stopping. (The rail company is obliged to provide you that footage, upon payment of a fee which can't exceed £10, under the provisions of the Data Protection Act). However simply inviting the police to check the footage for themselves (which wouldn't cost you a penny) would probably suffice anyway.