Hi there, I'm also from the UK, I work on a large organic garden. Even in a professional garden we don't have enough mulch. But I have found some great sources.
Nettles!find an area and cut them before they set seed. I maintain a river path and have an unlimited supply of very nitrogen rich mulch. You can cut several times a year.
Grass! Don't use the first cut of the year as it will have seeds in it. Offer to cut someone's lawn or a public area, so you are giving something back.
Both of these are good excuses for getting a scyth or sickle.
Weeds can be used. Nothing with seeds and make sure tap rooted plants have fully dried out.
I don't know your situation, but how about chickens? The manure is rich and straw bedding is great for composting.
Buy your friends and family compost bins and regularly collect it so they keep doing it. It's a good excuse to catch up.
Leaves! Find a local path or an area you can clean up. Leaf mulch is the worth the effort and collect way more than you need.
Grow more plants, it takes a while but once your garden is running, you will get a steady supply of remenants through the season.
If you are feeling ambitious about your garden, get comfry growing immediately!
I also don't drive, but I do have a bike trailer! Good luck