Which thickener should I use for fake spaghetti out of fruit juice?

I've seen it done with agar-agar on the Danish show "Spise med Price". They made spaghetti with lemon balm. They sucked the warm liquid with agar-agar in it up with a syringe, pushed the liquid into a thin plastic tube, which they lowered into ice water. Before they served it, they pushed the spaghetti out of the tube with the syringe.

As for a flavor idea, they served it with carrot cake made in mere minutes (from start to finish) in a microwave oven.

Check out this PDF called 'Texture: A Hydrocolloid Recipe Collection'. It has some recipes for various types of spaghetti using agar and other hydrocolloids. Since agar tends to dissolve under heat, it also has a recipe to make noodles with methyl cellulose which gels when heated.

They suggest using a syringe to make your strands of spaghetti. It may be time consuming, but I could see it working.

I like your ideas for flavors! Perhaps mushroom bisque/broth flavored noodles with a beef stroganoff sauce.

I can't help with the gelling, but to make the strands, consider using (making?) a chitarra : it's a frame with parallel wires -- you lay a sheet of pasta on top, then use a rolling pin to force it onto the wires, cutting the pasta into strands. This would allow you to make sheets of gel, rather than trying to form each strand individually.

Some quick searching suggests they can be bought from a gourmet cooking store in the US for $40. I don't know how hard it would be to find one in the UK. I've also seen things that look like multiple pizza cutters mounted so they can be adjusted in how far apart they are. (it looks like they're called an "adjustable dough divider" or "adjustable dough cutter", and they run between $22 and $200). There are also fixed blades mounted on a single handle, and looks like the term to use is "rolling pasta cutter", which are much more reasonably priced, but not as flexible in their use)

Starch gelatinization. Not sure if it'll work, but it won't melt at high temperatures. Might be worth experimenting with.

Regarding flavors, try the Matcha tea ...also for the color of course,

Sheet gelatin. Commercial bakeries use sheet gelatin to make large quantities of gelled foods such as jello. It's perfectly edible and neutrally flavored.