Sadly, there has been a lot of spin marketing on bamboo generally. In this case, the bamboo is actually fine - it's literally the plant. But the toothbrush bristles are plastic. A couple of manufacturers have been claiming for years that the bristles are 'Nylon 4' and compostable. That is simply not true. Unfortunately, unless you buy a bamboo toothbrush made with a hogs hair bristle, you can't buy a wholely and fully compostable or recyclable toothbrush.
Bamboo toothbrush handles are fully compostable. They are wood and are definitely a way better option than a plastic handled toothbrush. But 99.99999% of bamboo toothbrushes have non recyclable, non compostable plastic bristles.
Bamboo toothbrushes are a great example of the kind of choices we have to make every day when we are trying to live more sustainably. Life is pretty much made up of what we call ‘the least worst decision’. Good at one end and not so much at the other, but still better than the alternative.
Buy bamboo toothbrushes and cut off the head. Make sure you use the toothbrush till it's properly worn. If you do have plastic brushes - from manual or electric toothbrushes, recycle at a Terracycle hub like Biome.