Just a thought I had but it doesn't work.
Changed a few things to get a better view of what's happening.
When the voltage rises across the potentiometer this triggers the transistor which in turn activates the relay, switching the battery out of the charging circuit and switches the led in.
Just built it for real and testing it out with a 1A/20v supply I put together from a transformer and a rectifier bridge, represented as powering a lightbulb.
The reset switch isn't necessary when not using smoothed DC since the circuit resets 100 times per second.
