I know this may make me sound like a pedant but the word is "current" not "amperage"
I would also guess that you have been getting your units confused. The current will have been 0.59 A (which is the same as 590 mA).
It sounds like a typical Lithium Ion 2 cell charger. When the battery is low the current is limited. In this case it is limited to what the charger can give out. I don't know what device limited the current but it is correct that it is limited like that.
As the battery gets charged, the voltage rises. The voltage is limited to 4.2 V per cell, so 8.4 V for 2 cells. The current falls due to the nature of the battery. At some point, the current gets so low that the battery is considered charged and the power is cut off.
You have several batteries charging. I can't work out if you want a light to show when they have all stopped charging, or a light for each battery.
If you can measure the current at the 12V input that is better, because the devices after that will not mind if there is a bit of a voltage drop. The easiest way to detect current is to have a small resistor in series and measure the voltage drop on that resistor.
The circuit attatched is quite crude but simple. The transistor needs to be a PNP power transistor, as it has to be able to take the entire charging current through its base-emitter junction. The light will be on when current is above the limit.
R1 is set so that I * R1 = 0.7V where I is the current you want.
R2 limits the LED current. 1 k
hm: will be fine.
The LED can be just about any LED you want.