... No need to worry about the reverse voltage, the lamp goes out when the voltages either side are equal, so there is negligible reverse bias.
This not quite right I'm afraid, and could lead to a serious problem.
The ignition warning bulb has to allow current to flow BOTH ways and an LED will only work with current flowing in one direction. Why does it need to allow and illuminate with flow in both directions? This is because sometimes the alternator fails and OVER charges. I have seen this a number of times during my working life and even experienced it myself not long ago.
I have to disagree, the alternator voltage always has to be higher than that of the battery in order to charge it. If what you say is correct then the lamp would always be lit. In a machine-sensing alternator, the regulator and field winding is fed from the ignition warning light via a diode which is forward biased when the alternator is not charging and allows the lamp to light and is reverse biased when the alternator output exceeds that of the battery, extinguishing the lamp. I think that in the scenario you describe, the diode had failed short-circuit, lighting the lamp and causing the regulator to malfunction, or possibly the overcharging had blown the diode, but had the diode not failed the overcharging would have continued anyway without lighting the lamp.
Sorry but I know I'm right here Peter. The ignition warning light is basically a form of 'balance meter'. When the voltage is higher one side than the other then current flows from the higher potential to the lower potential and the bulb glows. The bigger the potential the brighter the bulb glows. But it must be able to flow both ways.
The bulb gets its feed on one side from the ignition switch, which in turn gets its feed from the battery. So one side has battery voltage. The other side of the bulb connects to the alternator and when the engine is not running this side is connected to earth so the bulb glows brightly. When the engine is started the earth is effectively cut, and the regulator connects the alternator voltage to that side of the bulb, AND it is charging the battery so the battery side of the bulb also has virtually the same voltage. The difference is so tiny that a bulb will not light up.* So with the same potential both sides of the bulb, it goes out. (Probably if you had an accurate enough meter connected you would see the tiny difference)
If the alternator stops charging or its charge rate drops below the correct figure, the potential is now higher on the ignition side as the battery at this stage is still 12v nominal (actually probably 12.8 to 13.0 volts for a battery that has just been charging) as the battery has such reserves that its voltage is slow to drop. The alternator side will have dropped immediately so the bulb starts to glow.
Now if the alternator starts overcharging, the alternator side increases immediately above 14.4v which is the normal maximum but the battery will still be at roughly 13.2 volts (a 12 volt battery has 6 cells with 2.2 volts each max.) and it won't change because its mass and reserves do not change quickly, so the potential difference is now the opposite way and the current flows 'backwards' and the bulb again starts to glow. I've seen alternators that have started over charging go over 18 volts but a battery will simply get so hot it will boil the fluid away and buckle the plates before its voltage has increased to match the overcharge voltage.
In the sixties and seventies the Lucas ACR alternators had Zener diodes fitted that simply blew if the voltage spiked such as when jump starting but also if the voltage climbed rapidly for any reason. This protected the rest of the alternator. We often used to fix the problem and simply replace the Zener diode and the alternator was fine again. Saved us a lot of work on those vehicles where it was a swine to replace the whole unit, but getting at the back was easy!
Roy
*P.S. this is another reason for using an incandescent bulb - it takes a certain amount of potential difference and current flow before there is enough to make the element glow.