"Biodegradable" is a broad term, and may not mean that the product will disappear on its own in a few days. For details on biodegradable and similar alternatives to plastics, see here.
The solution is to use loose tea. However, I've realised that many teas come only in tea bags!
Moral Fibres has a summary of different brands of tea bags that do not contain plastic - Teapigs, Twinings pyramid teabags, Pukka (each teabag is individually packaged in a plastic sachet though), Tetley (only string & tag teabags used for catering), PG Tips pyramid teabags.
Abel & Cole & Clipper are plastic free according to their own website.
The Independent has a few others to add to the list - Brew Tea Co, Hampsted Tea, We are Tea, Eteaket, Nemi, Roqberry, Good & Proper, T2.