We've just gotten some old books out of the loft, and it appears the plastic bags they were in have disintegrated/biodegraded.
What is the most environmentally friendly way to dispose of them?
Unless the plastic bags are labelled with a recycling symbol, there is no easy way to know if the plastic is even recyclable.
Good biodegradable plastic bags are made from decomposable materials, which contain no pollutants or toxins. The fact that your plastic bags have disintegrated doesn't mean they are biodegradable.
The safest thing for the recycling plant (and the environment) is to assume they aren't recyclable and dispose of them as non-recyclable waste.