You can claim that the oak is causing a nuisance and you want the branches pruned back beyond the boundary.
In default, you can prune back to the boundary but not beyond. The branches do belong to the council.
Any further questions or queries. Happy to help further.
If they are under a TPO then the council has discretion as to the extent of pruning. You cannot prune without permission.
You can make a formal application to the council. You need formal permission. You can check the procedure on www.planningportal.co.uk, click the do I need planning permission link, there's a list on the right, scroll down to trees and hedges and click the link, there you will find a link to the government's guidance notes and a link to make an application. In the guidance notes check flowchart 3.
The council has discretion as to its decision but you could appeal. There is no case law precedent that applies.
The cases I have been able to find relate to works carried out without permission or where a challenge to when the council refused permission. I have not found any report case similar to yours. That is not to say that there is a precedent but on the resources I have access to I have not found it.
My point was that a council would not have to take a decision in accordance with any previous decision, generally in planning decisions a council is not bound by previous decisions. It would be down to you to appeal if you do not agree.
It is always difficult to give any estimate of chances of success with any degree of certainty. Even when a precedent applies. Each case turns on its facts and can be distinguished from any precedent. If you constitute the overhanging branches to be a nuisance then you can make your application that the branches should be pruned back to 3-4 metres from the building line, you can incorporate a TPO survey that such pruning will not damage the protected tree. You have the right to appeal.
I am not a barrister and if you wish I can opt out on the chance that there is a barrister online who can assist.