They are quite sticky on their rules, Hairylee.
Perhaps you find out if you have a neighbour who doesn't fill there bins every 2 weeks ... NO, don't just do it ASK first. I invite my next door neighbours to do this (either side), and one has a young baby.
Was going to suggest to ask for a green bin, but can't put vegetable waste in there such as rabbit left overs, however, there is a new scheme coming into play in March .. think it means you can deposit this type of waste in them, bit ambiguous..
Recycling - Your Green Bin
Other than that, try flat packing the nappies (or similar). Maybe 2 at a time, inside to inside if that makes sense.
After that, afraid is a journey to the tip for you.