There are so many ways you can take the "sustainable" concept with respect to a ukulele, and I think it really just boils down to what market segment you're after and what your ultimate goals are.
Wood is challenging to recycle post-consumer, and has pretty limited re-use (you can recycle a recycled aluminum can, but you can't really process wood repeatedly into an equivalent product). So pretty much any wood-based ukulele is going to have to come from what is effectively virgin wood, even if it is waste from another wood process. bad news though, there is already a pretty significant pre-consumer "recycling" industry producing all kinds of wood products out of scrap or waste from other processes (i.e. making wood pellet fuel for pellet stoves from the waste sawdust produced by a hardwood flooring manufacturer). Those industries are far enough along that there can be significant competition for the waste material. On the other hand, ukuleles do have the advantage of being small enough that they can already be built with what basically counts as "scrap" for a lot of wood-based industries. You could probably run a business making ukuleles from the offcuts of a high volume cabinet or furniture shop. No need to process the scraps into a wood-like product.
Ultimately though, wood may not be the best resource if you're aiming for sustainability or eco-friendly goals, since it is pretty difficult to replace at the same rate as it is being used.
Luckily there are already a few pretty well established wood-like materials that get around the problem with how long it takes to replace trees - i.e. bamboo (which can be processed and used as if it were wood, per the uke pictured in this thread already - or it can be shredded and processed into a composite "board" similar to what you seem to be after), or linen (which can be processed into a fiber that can be made into a composite plastic with similar properties to carbon fiber or fiberglass). You commented that you're not sure if the material you're after exists yet or not - if you're able to include these sort of "wood alternative" materials, I think we're already there.
That just leaves the problem with market acceptance. If people bought ukuleles purely on utility (i.e. tone/volume per dollar spent) I think it's pretty clear that the industry would look very different than it does today. But people don't just shop with their ears, they shop with there eyes, hands, emotions, and their sense of tradition or fitness for a specific style, culture, or artistic expression. You've indicated that you think it'll be a challenge to come up with a material that can sound like a tonewood. I really don't think that's the case - I think the much more significant challenge is coming up with a material that people will actually want to buy, regardless of what it sounds like.