a tree with a TPO on it CAN be trimmed or removed without permission if it is dead, dying diseased or dangerous, but it is sensible to either get a tree surgeon's advice, or to get some pictorial proof before acting. I have had this situation twice now. Smurf if the tree is damaging your fence then you might like to drop a note in pointing out that the owner of the tree is liable for all damage caused by the tree regardless of whether or not the tree is the subject of a preservation order. You culd also put in a bill for fence repairs to date. I don't think there is any legal action that you can take until after the damage has occurred.
If it was me and the neighbour had refused to prune it, i would be checking on its preservation status (call the council) and then if ok I would be paying for someone to come in and prune off the stuff that overhangs your garden. I would be viewing it as part of the costs of maintaining the house generally. trees with a TPO can be pruned with permission and you don't have to be the tree owner to apply for and receive that permission.