So what grade of fibre is used in these displays? Clearly, it can't be terribly expensive. What are the limitations of this type of fibre?
I think I have found what I am looking for.
PMMA Optical Fibre End Glow seems to be the standard term. PMMA being the plastic material of the Plastic Optical Fibre (POF). "End Glow" refers to the end being the light output, as opposed to side glow which is also used in optical fibre lighting.
It is available in multiple widths (0.5mm, 0.75mm, 1.0mm, 1.5mm, ... , even up to 6mm), sheathed, unsheathed and sheathed with multiple strands.
Looks like it can be bought from Alibaba or eBay (it looks like fishing line) or from lighting installers such as this one.
Not having looked at these closely I would surmise that the material used would be more accurately called a light pipe. Fibre optics with guided light is designed to ensure that the light gets to the other end and even the least expensive version (used in GPON) can go many 10's of metres. These seem to emit light along a very short length.
I would also suspect that these light pipes just use a single material in stead of multiple layered or variable index materials to keep costs way down.
PMMA, or fishing line if you don't care about the loss. Large diameter for easier coupling of the light into the fiber. You can use a Razor Blade to cut it but make sure it you cut it straight across with little to no angle.
You could also just bundle it, wrap with black tape and put a light under it. Use an LCD so you don't get the ends hot.