While I'm not suggesting that any parent should actually allow a really young child to travel unaccompanied, the answer to your question is 5 years old.
That's because under-5's travel free but have to be accompanied by a fare-paying passenger. 5-year-olds pay half-fare and, as fare paying passengers, they're (at least theoretically) permitted to travel on their own.
In many parts of southern England (especially in the wealthier areas where the majority of children attend private schools), thousands of children of primary school age travel daily from home to school by train, without accompanying adults. (Mum or Dad probably takes them to the station. The children usually travel in groups, which are then met by the school minibus or coach at the other end of the rail journey, but there's often no adult on the train with them).
Chris