Just to weigh in on this one...
There are a few points to make:
1. Light: You do have a right to light at your property. In fact, your right to light is protected under Rights of Light Act 1959. However, trees and light is a tricky one because trees do not block all light, as they shed leaves etc. therefore, any case made about trees blocking light is thrown out, despite the fact that trees can block light during the leafy seasons. Additionally, because there has been no case law since this act to support the removal of a tree due to blocking light, there is now a general consensus that trees are not seen to significantly block light in the eyes of the law. Now, this brings us on to TPOs. Because a TPO is legally enforceable under the Town and Country Planning Act, no local authority will grant works justified by light - its just not going to happen. Hypothetically, if a LA did grant tree works due to light, the flood gates will open as this then established a precedent where there are now some circumstances where works will be granted for light reasons. No tree officer in they right mind would allow this to happen.
2. Damage to buildings: I come across this one all the time, I have considered applications to remove trees because of "roots under the foundations causing damage / potential to cause damage." It is quite straight forward, unless you have a report from an industry professional (usually a structural engineer or building surveyor) to suggest tree related damage, your application will not be considered. Moreover, after spending years in the subsidence consultancy industry, the most common type of damage to buildings caused by trees is subsidence, which is known as indirect damage. Direct damage (i.e. direct root expansion against another object) is SO rare in low rise buildings. It is more common on boundary walls and made ground (driveways or paths for example). In these cases the tree could be removed (or not) depending on the circumstance. Though, if a large TPO'd tree was causing damage it is more likely that the driveway will receive works and not the tree. Most local authorities use Capital Asset Value of Amenity Trees (CAVAT) to determine cost effectiveness. If your property is getting damaged by a 3rd party tree, claim on your buildings insurance.
3. Foreseeability: it was mentioned in this thread about the tree in question potentially damaging the house or a branch falling and damaging property. It was also mentioned that the local authority would be liable in this instance. This is partially correct, the liability is with the tree owner. They are responsible for the maintenance of their trees under their duty of care. Usually this would mean that the tree is surveyed every 1,3,5 years as required. Even then, if the tree did damage the property and it was not foreseeable, the tree owner would not be liable as in the eyes of the law they would have taken reasonably practicable measures to ensure the safety of their trees. I have had this several times with my trees where some minor failures have occurred in high winds. The claimant party claims on their insurance, our insurer requests our surveys to evidence our compliance with duty of care, we produce 10+ years of survey records, the claim is denied as we have taken reasonable steps and the tree damage / failure was not foreseeable. Therefore, if there is no foreseeability and the duty of care is complied with, there is no claim. Though, if the damage / failure is foreseeable and the Local Authority deny an application to fix the issue, you can evidence your efforts and there is a case to pass on liability to the Local Authority.
Regarding a possible solution: If the tree canopy is within 3 metres of your property, apply for a 3 metre clearance, this will likely be approved and help with your light issues. Though, regarding the damage to your property, if you deem it significant enough, log a claim with your building insurer.
I hope this provides clarity.