I have an old leather coat which over the years is getting more and more scuffed: the black is fading but the leather is still okay underneath.
How can I get it back to a (more or less) pristine state?
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Test this on a part of the coat that is not visible in normal use, for colour matching as well as to see that you have the right product.
Bring on a good coat of shoe polish, of the kind that has been around for many years, which is put on with a cloth or small brush and after a while to get into the leather, brushed out with a clean (bigger) brush.
Mostly you can get the better kind in shoe shops or places where they repair shoes. It has to be a container with a fat, saturated with the colour you want to restore in the coat. Not the sponges with 'quick fix' shoe polish.
For the first 'repair' best bring the polish up in a fair layer, give it time to get into the leather but do not leave it for hours or even days, and wipe it off with a cloth before you use the brush. Or do not use the brush this first time but wait till you have done it a second time. Brush lightly and keep an eye on the leather that you do not damage it.
Repeat this a few times, so the leather gets a decent protection and keep repeating every few weeks (or months if you do not use the coat often.) Always do the shoe cream routine before you store the coat for the season.
From the way the question is worded, I think the OP really want to improve the looks of this coat. But others may want to keep their leather scruffy and use it as it is, not using anything to improve the looks.
A coat that has become scruffy in a certain way shows age, just like wood use over many years shows patina. If it is good looking, use it as is.