Best mulch for veggie garden?

Hi all. I'm building some new veggie beds, using the "lasagna gardening" method which consists of basically building a little compost heap right where you plant your veggies. My friend has used this method with great success and I'm unable to afford to purchase a ton of soil, so figured I'd create my own!

I'm new to the UK, from Australia, and need your help regarding which mulch (brown-matter) to use. I was looking for pea straw, lucerne, or sugarcane mulch. But after an hour of searching, frustratingly, it seems that the only appropriate mulches you can buy around here are woodchips - which I understand are NOT good for my purpose as they'll leach nitrogen from the soil - or straw, which I remember learning often contains lots of weed seeds. As I'll essentially be using this brown matter as a component of a compost pile, I will need it to break down rather quickly which is another reason woodchips wouldn't work.

Unfortunately we don't have any piles of leaves around or I'd happily use those. This year I'll have to buy something in. What would you use?

Thanks in advance for your help!