I’m writing this just in case anyone else has this problem, as I finally found the answer after many hours of frustration.
When I first upgraded to Blender 2.81 and loaded my current project into it, almost all textures were low in saturation, both in the Viewport and rendered.
Eventually, I discovered the problem. In 2.79, I had duplicated most of the image texture nodes and set the duplicate to “non-color data” for use in bump-maps. This seems to have been fairly common practice as I’ve seen it in at least 2 tutorials.
Anyway, for some reason, 2.81 had set all of those textures to “non-color”. I also noticed that 2.81 won’t allow the same texture to be set to different color-spaces. As soon as I changed one node to sRGB, the duplicate also changed.
After changing all the image texture nodes to sRGB, the colours came back. The bump-maps also seem fine, but I could use hue/saturation and brightness contrast nodes for more control.
Anyway, I hope this might help someone else, as I couldn’t find another reference despite a lot of searching.