Sighs. I suppose it was me that started us off on this martial course. What else should we be looking at in terms of inconveniences?
Horseshoes (and horseshoe nails) for a start. So much a part of a smith's work that he was considered as a farrier anywhere outside cities. I can't think of a convenient alternative, so horses would probably never be shod, and roads wouldn't be stone surfaced, or else equine traffic would be much reduced. Donkeys and oxen for freight, but not the same chivalrous upper class. Scythes, sickles – we need them. I suppose glass, but hit a rock – and after you've ploughed – assuming you can plough; getting a decent ploughshare without iron is complicated. Either a primitive 'wooden stick scraping the dirt (not anything like as effective as a true blade breaking up the soil), a bronze blade (too expensive, and wears too fast) or maybe ceramic, but I'm not that convinced) and the tendency to shatter – bone or antler seem too soft, won't hold an edge againstthat sort of work.
I'm wondering if you could cut grain with a leather whip; be tiring, but perhaps. Or a machine with wooden scissor blades, edged with something abrasive, needing frequent replacement but easy to manufacture.
Wood might be usable in the form of plywood; glues available to the Middle Ages were quite good enough for this purpose even leaving aside the possibility of using a rubber-based cement for the laminating.
Mediaeval adhesives weren't really very good; essentially size based (size being boiled down animal leftovers, like hooves and bones, not a dimension) or resin based (being tree extrudence); not very waterproof, or all that permanent). Indeed they've improved massively during my lifetime. I would not choose them as an essential component in either body armour or a shield. However, if invention in metallurgy was blocked, it's quite possible organic chemistry would have advanced faster.
If you were building laminated armour, you wouldn't start with a sheet of plywood and steam deform it to shape (which is basically how to build a mosquito) but make a mould of your warrior's chest (or whatever) and press your softened strips of wood into it, impregnating with adhesive. You can laminate in leather, or horn, or even ceramic or stone while doing so.
The trouble with wooden or fibrous armour (fibreglass could be built pretty early) is that rather than getting dented (which can be hammered out, making the armour reusable) weapons dig in to the softer (not soft, just less rigid) material, and it is slowly weakened, and is far more difficult to repair. (which hasn't prevented armour from bamboo, various woods, crocodile hide, splint and scale… all of which have similar difficulties.