Sadly I think you are wrong. Public liability applies here if neglect or negligence occurs, the point about storm damage and similar acts of god is that no neglect or negligence occured.
You can disagree but the OPs insurer knows this, I've understood it for a while which is why I raised it early on, others have known it and a google search also support it with all the cases I've read to double check
Viberts is a full service law firm. We support our clients with outstanding legal advice and customer service. We are with you every step of the way.
www.viberts.com
which includes the paragraph..
"In fact this is a common misconception. Your neighbour will not be liable for the damage caused to your property as no-one is held responsible for damage, such as storm damage, caused by so called ‘Acts of God’. Generally you should look to your own insurance company to cover the costs of removing the tree and repairing the damage. Similarly, if a tile blew from the roof of your property and caused damage to, for example, your neighbour’s car, he would have to claim against his own motor insurance."
Feel free to google and find an example where such a circumstance occured as you describe as I'd be interested to see it.
Not withstanding this, and I've mentioned it a couple of times, the neighbours insurer may include cover for such a situation, but thats a policy and not legal decision.
Please don't shoot the messenger