In the UK this type of activity is generally carried out by the local authority. Green waste is collected weekly or bi-weekly via dedicated bins supplied to individual households, and the service is usually included as part of local property taxes. Sometimes a small extra fee is charged.
As to what happens to the waste...
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Once your food and garden waste is collected for recycling, it’s taken to a composting facility. This is a way of composting all types of food and garden waste in an enclosed unit. The mixture will be shredded and put into special tunnels where air circulation, temperature and moisture can be controlled to break down the food and garden waste into soil conditioner. It is then used as a rich soil improver called ‘Pro-Grow’ and can be bought at Household Waste Recycling Centres or via this page.
So ultimately they sell it back to you, probably at a loss.