I’m not sure where you’re located in the world, but in Europe Dyson utilise the WEEE recycle scheme. They recycle older machines within 28-days of purchase. This only counts for the machines however. All spares, accessories or other components, like filters, should be recycled locally.
Across twitter and Facebook, they’ve previously recommended checking with the local recycling centre. They may suggest placing it in the home recycling or taking it to a central location.
I replaced my mother when I went back home (State side). We called the local municipality, they suggested we disposed of it it at home.
As it’s not got any electrical components, you’ll be fine using ether a local recycling centre, depot or your own recycling bin. There’s no postal address or service for recycling directly with Dyson.