You don't have to inform the police in order to claim - the accident didn't occur on a public highway, and you weren't driving! Contact your neighbour's insurance company and explain the situation - if they are any good, they'll arrange for a garage to collect your car and arrange for an estimate/repairs. The only problem you may encounter is if your MOT has also expired, and the damage is enough to write the car off - then you could expect a reduced offer.