I assume here that you want clothing designed for cycling, or at least for sports, as much as possible, so for significant commuting, leisure, or fast rides, but not just wandering around town, as that could be done in everyday clothes. I deliberately haven't linked to products, as I don't want to imply any endorsement. I've also mentioned clothing for other areas, as I came across things while I was looking into your main question and they're useful for the broader underlying question.
Merino wool is well known in cycling and other outdoor activities, especially as a base layer and for socks. You can buy jerseys quite easily but there's not much for your lower half
Bamboo fibre is almost always rayon, almost pure plant cellulose. Bike shorts and many other forms of fitness clothing are available. As cellulose, it should be biodegradable (over long periods I suspect) and it's from potentially sustainable sources. Other forms of rayon may also be suitable. I've had socks made from it in the past; though not specifically for cycling they seemed reasonable even when my feet got wet.
Gloves may be tricky. Leather is your best bet to keep the wind off and provide some protection from water, perhaps oversized to go over wool in the winter. Suede is good for gripping. Any padding won't meet your requirements - so comfort may be an issue.
A waterproof top layer might be tricky, unless you go for a waxed cotton jacket, but they tend to be bulky and heavy, and not designed for cycling.
For shoes, you'll also struggle. Leather bike shoes exist but are uncommon and by the time leather has been tanned using modern methods it's almost a petrochemical product held together with a bit of cow. Soles etc. are universally synthetic. If you really want biodegradable shoes you can probably get some espadrilles with cotton uppers - they'd even look quite stylish on a vintage Pashley, but you're not going to get cleats on them.
However if you're worried about microplastic pollution, shoes are much less of an issue as you don't wash them very often, and washing is when they shed. This also applies to waterproofs. When you've completely worn something out, you can dispose of it properly (yes, landfill, but a lot of textiles end up there whatever they're made of).