This isn’t a human rights issue.
It is a common law nuisance and negligence issue.
The tree could be problematical for 2 reasons firstly if there is anything wrong with the tree or if it blew down, and it damage to your property and secondly, if the roots are potentially damaging the foundations. The roots will generally spread out as far as the tree is high.
There is no easy way of appealing the Tree Preservation Order once it is in place.
You have probably seen this website from the government
firstly, you have to ask for consent to remove the tree and then if they refuse, you appeal.
The grounds for appeal are on the form but what I would also suggest you do is make the application for consent and with it, put a covering letter advising the council that if the tree causes any damage to your property as a result of their failure to agree to have it removed, the you are putting them on notice that you would be seeking compensation from them.
That letter would be better coming from a solicitor or planning consultant. If you want to have the maximum chance of getting the initial application and failing that, any subsequent appeal through,I would suggest that you use the services of a planning consultant who will know exactly who the personalities are in the local authority planning department and how they work and will know exactly what’s in their minds and hence, out to word the initial application and any subsequent appeal.
Can I clarify anything for you?
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We can still exchange emails. Best wishes. FES.