Gauze tape, steristrips, or little bandage rolls.
They're cheap and help, trust me.
I once cut myself pretty bad while dicing onions (I was out of practice and had just sharpened my knife for a good while, pretty damn sharp. We're talking anime sharp.) and had to go to A and E a few nights before my exam. The cut was about an inch deep into my left thumb.
If I had some antiseptic and gauze/etc, I'd have been fine.
So, if you have gauze, little strips, some bandages, and antiseptic, you won't need to go to A and E.