If your compost material is noticeable wet it is too wet and that is why it is composting "rather slow". Any new material you add now should be shredded into somewhat small pieces and added dry, not wet. If air seems to be a problem that will be necause of the wet material. Moisture excludes air.

The veg & grass greens might have had a high water content. 1 thing you might try in the future is to dry the lawn clippings a bit, spread them out for a couple of sunny days (good use for the cardboard). You'll know if your grass is too wet if it clumps up in the compost, this means it isnt mixing with the browns.
Unfortunately I find that paper shredded into 1/4" strips do take a bit of time to compost (the cheap paper.shredders produce this size, the $200 models offer the micro-fine shredding!) I resort to paper when I run out of leaves. I think shredding is better for composting, but realize that chore can be a pain, so it depends on how much (or how little) time to produce compost you want.
i find it easier to add water to a dry pile than to contend with a pile too wet. a test is to take a handful of mixed material, does it feel wet? If squeezed, a couple drops of water is OK, more means too wet.

My shredder would handle thin cardboard, like a cereal box, but not a heavy corrugated cardboard. I wet that first, then it's easier to tear. If your pile is too wet, you could let it dry before adding to the bin.

Karen

Crumpled cardboard will usually degrade slower than shredded (although matted layers of paper products can resist decomposition pretty well) but it will prop up little air pockets in your pile which might help the rest of your pile to break down quicker. Perhaps there are also factors to UK climate that also make a difference for you but that do not apply as much in other places?

Sorry if this is off-topic, but it is regarding cardboard. Are pizza boxes, cereal boxes, and toilet paper rolls OK for composting and for use on vegetables? Seems there is glue in the toilet paper rolls and beyond the seam of cereal boxes (that I always rip off as it's obvious there is glue to hold the box together), it seems the glossy paper is glued to any cardboard material. Is that glue broken down during composting enough to use on vegetables or should I only compost cardboard that is solid brown?

No cardboard is safe here, all gets put through the cross cut shredder and put into the compost. If you think about it, would they use something unsafe on something that holds your food? I hope not.
Here everything turns into rich compost in no time.
Sandy

I never thought shredders were worth the time until my wife FORCED me to get one. Now I'm quite impressed. My pile has gotten to a size where shredded paper and cardboard are pretty much unrecognizable in about two weeks. Torn strips of the same stuff last almost twice as long. It seems that having increased surface area by shredding the paper into small pieces has made it quicker to decompose.

Curt

brdldystlu, why would it matter for the box to be safe? The food is isolated from that usually in a plastic bag. I am still leery of putting in anything like the seams that are obviously glued together. I tear those parts off. My main concern is the glossy picture part of a cereal box or the like as that glossy part is actually paper glued onto the cardboard. Im sure thats not some environmentally friendly glue they use as there would be no reason for them to do so.

The big reason I can think of for glue to be environmentally friendly is because its cheap.
AFAIK most basic paper and wood glue is still made the old fashioned way from, uhh, renewable resources if you get my drift. Very little in the farm economy is wasted.
All regular glue, like Elmers etc. is non-toxic so the kiddies can eat it. I have no reason to believe that packaging glue is anything different.

Corrugated board (cardboard) is made from paper and cornstarch adhesives. Most of the cardboard in the world is made in the USA. The glue is just one of the hundreds of products made from corn.

I find that cardboard packaging composts quickly. I line the bottom of my compost bin with a heavy layer of CB to keep the tree roots out. It only lasts about a month.

Ironically, the US sends ship loads of corrugated boxes to China which are then filled with Wal*Mart goods and shipped back.

Thanks for all the kind advice. Looks like I need to buy a shredder. Pending that, I put a bag of straw from the pet shop in, mixed it all up and it's COOKING! yay!

If you get it wet, you can rip it into reasonable strips or pieces by hand rather easily. When it's dry- it's a pain to do any volume. This doesn't shred as fine as a shredded, but it'll work rather well anyway. As my class-act dad would say: "Good enough for the broads we hang out with". :)