This is potentially called the tort of nuisance
You need to write and request they cut the tree back within 14 days or say you will go to Court within 14 days. You should make sure you send this recorded delivery and keep a copy.
If they do not then you can issue proceedings in the County Court. You can either do this by completing form N1 http://hmctsformfinder.justice.gov.uk/courtfinder/forms/n001-eng.pdf and take it to your local County Court.
You are asking the court for an order that they cut back/prune the tree
The Court will then issue a claim which a copy will be sent to the Defendant who will have a limited time to defend it, if not you can enter Judgment and enforce.
If the matter is defended it will be set down for a trial. If the claim is allocated to the small claims track you wont need representation. If the claim is allocated to another track then you will need representation for the final hearing.
You wont get legal aid for this but if you win then you can ask the Court for your costs back
Can I clarify anything for you about this today please?