You would normally need to apply for permission from your local authority to cut back the branches, see here for further information: https://www.gov.uk/guidance/tree-preservation-orders-and-trees-in-conservation-areas
You will see from the above link that there is an exception for work to prevent or abate a nuisance as follows:
"What is the exception for work to prevent or abate a nuisance?
The authority’s consent is not required for carrying out the minimum of work on a tree protected by an Order that is necessary to prevent or abate a nuisance. Here ‘nuisance’ is used in its legal sense, not its general sense. The courts have held that this means the nuisance must be actionable in law – where it is causing, or there is an immediate risk of it causing, actual damage.
When deciding what is necessary to prevent or abate a nuisance, tree owners and, where applicable, their neighbours and local authorities, should consider whether steps other than tree work might be taken. For example, there may be engineering solutions for structural damage to buildings."
If in doubt, you should speak to your local authority first.
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