There are several oak trees on the land surrounding our property and they're a nightmare. I really wish I'd been more aware of the problems before we'd moved in. Caveat emptor and all that.
The house had an extension added around 15 years before we moved in. It was a self-build job and I suspect not entirely compliant with building regs. We have had problems with heave in the extension, which were largely down to the nearby oak tree (with oak trees, the roots are far more of a problem than the overhanging branches). The solution, unfortunately, was for the tree to come down - our insurance company insisted on it. The TPO was irrelevant. Fortunately, the insurance paid for the job and our neighbours on whose land the tree stood didn't kick up too much fuss because they ended up with a job lot of high quality oak wood to sell.
My feeling is that there need to be much tighter controls on building properties near ancient trees that are likely to cause such problems. For me, the proper solution to this situation would have been for the house to come down, not the tree that predated the house by several hundred years. Though of course I'm somewhat relieved that we still have a place to live in.
I will never again buy a house with oak trees within a couple of hundred metres and would advise anyone else thinking of doing so to think again. Willows can also be extremely problematic - we also used to have a willow tree that had to come down because it was interfering with the drains.
Then there's the acorns and leaves... our driveway is always practically knee-deep in the feckers at this time of year.
