I bought my '98 VP in October 2017. The previous owner pointed out how the Battery Not Charging message and red light are always on, and that a new alternator would probably fix it. It has always come on when I start the car.
I put a new battery in pretty early on, January 2018. I was always keeping an eye out for any warning sign that the alternator was bad, but nothing like that ever popped out at me.
In August 2018, I took it to an independent Jaguar mechanic and told them about the light. Their first guess was that it was a bad diode within the alternator, so they put a whole brand new alternator in, but the message still came on. They put the old one back in, and load tested it, and said that it was fine. They told me that it could be coming from a loose connection somewhere within the dash. And putting a new dash in, or rebuilding it, could possibly make it go away.
Sometime last year, I left it running for a minute or two while I ran into the house to get something. When I got back in, I looked and for the first time that I'd ever seen, the message and the light were completely off. But then I started driving, and a few minutes later, they came back on.
It did this a few more times, either while I was driving or while it was idling for a little bit. I couldn't predict when it would turn off, and it would always come back on. I'd say that at least 99.9% of the time that the car has been running, the light has been on...but not 100%.
It turned off again the other day, for the first time in months. I was driving down the freeway and it kept flickering off and then back on, immediately back and forth. When I got into town and then home, it stayed on, and has been on again since.

It's just strange. I bought the car at 99k miles, and it has been through quite a bit - a transmission rebuild, among other things - but this has always gone on. without any clue as to what it is. I know that the Battery Not Charging light coming on is a fairly common issue on these cars, but any cases that are similar to mine? Or, any guesses on what it could be, or anything that I could try doing?

 

I think you may already have said what it could be and I ask, ...have you pulled out the dash to look at your connections? Hoping it will somehow go away isn't a fix.
If someone has been in there and not properly seated something, it needs attention.

Have you checked your battery terminals? Any corrosion on your battery cables can cause issues. If you have corrosion, replace the cable, you won't get it out.
Check your grounds to include the grounding lanyard from the battery, it can go bad (I can attest to that).

 

Another location that will cause problems is the bulkhead connector that carries current from the trunk to the engine compartment. It is on the right hand side and is connected to the cable going to the starter. Make sure it is corrosion free and tight.

 

First, remove the gauge cluster covers (its a pretty simple process) so you can get to the two connectors going to the gauges, and give them a jiggle to make sure they're not loose. You can even do this while the car's running to see if it'll affect your battery light. All the battery light is, is a signal wire going from the alternator vreg to the dashboard. If those are fine, and your alt is fine, and your voltage readings while the car is running are fine, then there's probably an intermittent break in the wire somewhere between the alt vreg connector and your dash. Downloading the electrical guide PDF for your year car will be very useful.