After hearing the horror story of another member's E39 overheating, I couldn't drive Jane without staring at the coolant temp gauge. After realising that I would get the same utility from staring at my left thumb, I decided it was best to do something about the ridiculously buffered temp gauge that comes in these cars. With the advice of @Allanw, I bought a cheap OBD2 scanning tool, and set about adjusting the temp gauge ranges.
This was actually a lot harder than I anticipated. It took me about an hour to get all of the software downloaded onto my laptop (user error), and a lot of the forums that have discussed re-coding these gauges were written from an E46 perspective. In my naivety, I assumed that since my E39 shares the same M54 as the E46, the process would be identical. It turns out that the coding side of things is completely different. This took another long while of internet trawling before I was finally able to get it done.
As a result, instead of my temp gauge telling me I'm at 'normal operating temperature' from about 75C to 115C, it is now reduced to roughly 85C to 95C. One of the best mods I've done for my mental stability (and for the engine's sake of course), and highly recommend it for anyone with this era of BMW (E38, E39, E46 etc.).
If anyone wants any help with this I am more than happy to help. Increased accuracy of our temp gauges is objectively a good thing; it gives the driver more notice when their car may be overheating or running too cold, AND hopefully prevents people booting their cars before they're actually at 'normal operating temperature'. More accuracy = greater chance of survival of these engines/cars and I am all for that!
I also checked for error messages, and was pleasantly surprised by the lack of any current fault codes! She treats me well, this old girl does.