Responsibility for most watercourses lies with the the riparian owner, the owner of the land on the bank. What follows has kindly been supplied by the Environment Agency (slightly adapted):
If you own land or property alongside a river or other watercourse, then it is likely you will have rights and responsibilities as a riverside property owner. In legal terms you are a ‘riparian owner’. Some of your responsibilities include:
• maintaining river beds and banks
• allowing the flow of water to pass without obstruction
Maintenance of the section of river falls to the riparian owner and they should always be contacted in the first instance. The only exception to this would be larger navigable channels where the Environment Agency has a responsibility to provide navigation and would be likely to clear a blockage. On all channels designated main river* the Environment Agency holds permissive powers to complete maintenance where it would be beneficial for flood risk purposes, depending on finances available.
All works within 8 metres of a main river require a Flood Defence Consent. You can apply for a consent by contacting firstname.lastname@example.org and discuss your proposals before submitting a consent. Consents are issued by the Environment Agency and are used to ensure that no works on or near a river increase flood risk to other people. A Flood Risk Assessment may be required, depending on the complexity of the works.
*Main river: these are usually larger streams and rivers. In England, DEFRA decides which they are.
For more information visit https://www.gov.uk/guidance/owning-a-watercourse