We love bamboo and were looking for a toilet roll holder in bamboo, but could not find one.
IKEA has Dragan which is a set of two bamboo boxes with lids. The larger one, when upright, is just wide enough to enclose a roll of toilet paper.
I cut of a length of a broomstick on which the toiler roll rolls. On both sides I stuck in a nail with a large head, possibly carpet nails, not sure where I got these from, just had them in a box.
As you need to be able to lift the new toilet roll into the box avoiding jamming it to the top, aim at having the incisions for the nails support to be below the middle of the height of the box.
I made the incision the size of the thickness of the nails, allowing the roll to slide inward and then downward. I did this with a power jigsaw. Be careful: bamboo is quite hard a material. Sand down the rough edges.