Has it done this ever since you changed the alternator Andy, or is this something that has just started?
When you switch on the ignition, 12V is supplied to the warning light from the battery via the ignition switch, the other side of the light connects to the alternator. Before you start the engine the alternator is stationary so not giving any charge - the warning light earths through the voltage regulator in the alternator & lights up. When you start the engine, the alternator also feeds 12V to the warning bulb so the bulb gets 12V from both sides, thus it has no earth & goes out.
Normally if the light does not come on when you turn on the ignition, I would check the bulb first. However, you say it glows when running therefore the bulb itself must be OK. So, either it is not getting the 12V from battery when you turn on, or it is not earthing via the alternator.
You mention that sometimes when running the voltage reads quite low, & the light glows, but then the voltage will rise & the light goes out.
Check all the connections at the alternator are clean & tight. Check the battery connections. If all this is OK then since the warning lamp earths through the voltage regulator in the alternator, I would suspect the regulator may not be working properly, which could also account for the variation that you are seeing in the voltmeter.
Tony
<font size="1">1981 TR7 FHC Cavalry Blue
1980 TR7V8 DHC Jaguar Regency Red - sadly sold!
1977 TR8 FHC EFI Factory development car Inca Yellow</font id="size1">