Coral, go on your local authority website and find the 'contact us' page. Fill out the 'general enquiry' email section, giving details of the tree in question and ask for the environment officer that has responsibility for trees to advise you on the best way to deal with the overhanging branches.
The local authority will (or should!) have published service standards so you will know how quickly you should receive a full response.
The 'tree man' will have a statutory responsibility to ensure any protected tree is not inappropriately lopped and will act/advise to prevent any damage to the tree as per the details of any TPO (Tree Preservation Order).
If the tree is causing any statutory nuisance, the council can order the landowner to take action as necessary. If the landowner is your landlord, they will then have to take ownership of dealing with the issue.
You gave the tree issue as an example of the ineffectiveness of your HO but stressed that he is 'unhelpful' with other issues you have raised. Next time you raise a tenancy or community safety issue with your housing officer, make a note of the exchange, including dates etc and details of your concerns as well as his response to them. Tell him that you anticipate a response within a specified time (check the landlord's service standards and pin this deadline to that criteria). Then write to the housing officer confirming the details.
If you still fail to receive a satisfactory response, utilise the landlord's complaints procedure. Include copies of the details of the initial contact as well as a copy of the letter of confirmation.