Cushions that are still in good condition may be reused by local animal charities or vets. Alternatively something like an old cushion can make an excellent kneeling pad. They may be suitable for reuse through a local reuse network such as Freecycle or Freegle. There are now a large number of reuse groups on Facebook.
Old cushion covers that are in good condition might be accepted by local charity shops. These could also be included in a clothing bank at a local Household Recycling Centre.
Otherwise these sorts of items need to be included in your black kerbside container or a waste to landfill container at your local Household Recycling Centre.