The old applying for abandoned car V5 but with a twist

You can only charge storage if you have sent the person a letter (or email or text message of you can prove that was their preferred method of communication) stating what the storage fees are and the date they commence. you also have to give a notification period, can’t remember offhand what it is but it’s something like 7 working days.

The vehicle can’t become yours until the storage fees have exceed the fair market value of the vehicle- and if it went to court you’d have to prove that you’ve assessed that value correctly and not lowballed the owner. Very difficult to assess on a ‘broken’ car and without the other party claiming you have a vested interest in undervaluing it. Ideally you’d need a third party valuation from a reputable source.

You also then take on an assumed duty of care to that property for the time period it is in your possession,,so if the situation switches and they turn up to pay off the fees but the car has storage damage that wasn’t there when they dropped it off, you become liable for that.
You also don’t have a right to withhold property, so if they turn up and want to take it without paying fees, they can call the police if you prevent them doing so. The correct recourse is to let them take it then persue the fees through the courts, but then you have to prove that they have cost you money by taking up your space.

These factors are why you rarely see anyone want to get involved with it, or see many people successfully do it.