If your citrcumstances are as worrying as you describe them to be, you have my sympathy. I am based in Scotland (not far from you) and I will add my say for what it's worth.
Firstly I think Wolfie's advice is generally the right tack to take but with a couple of refinements I would suggest.
The first is that the Occupier's Liability Act strictly speaking would not apply here. But the Act (and the scottish versions) are reallly just statements of the common law, they don't change the common law but they make proof easier. The Acts apply to harm or damage to you while visiting someone or while on their land. That isn't your situation, you are worried about damage or harm to you and your property from a tree on someone else's land that could fall into your land. Fortunately and quite sensibly the legal duties on the tree's owner are fundamentally the same as under the Acts. But just to summarise that bit, there is no statutory duty (i.e. ACt of parliament or the scottish assembly), but there is a clear common law duty.
Second thing is that in Scotland the ability of local authority to intervene in such a situation is significantly less than in England/Wales, as we do not have teir Local Government (Miscellaneous Provisions) Act. In effect in circumstances like this the Council can't do anything.
So you are left with common law. What does it say? It basically says that the tree's owner has a duty to ensure that his tree does not cause harm or damage to a neighbour. Two things to note right away. He only has to act reasonably if he should foresee harm or damage. Not just cut down all his trees in case they harm someone. Also this is a duty, not an obligation. The effect of that is not that he is guaranteeing your safety, just that if there is harm or damage his actions and inactions that could have prevented the harm or damage will be examined closely. In effect he cannot be compelled to do anything, perhaps until it is too late.
It would indeed be a good idea to have the tree inspected from a distance, on your behalf. The report will be of limited usefulness, as many aspects of the tree and perhaps one or more of its sides cannot be seen. The report could be sent (recorded delivery if required) to the owner, and to the Council if you wish as a courtesy. The report may give you some reassurance that the tree shows no outward signs of imminent failure. Even if the owner is left out of it, this might be enough to allay your own concerns. But if the remote inspection presents firm evidence of unacceptable risk, the report could also specify the works that would be required to reduce the risk to an acceptable or tolerable level. The ball will then be in the owner's court. He can have the implications for duty of care and for his insurance and perhaps his personal sequestration if he cannot meet an uninsured claim for harm or damage. But it is still his choice.
I suspect therefore that if the report concludes that even with limited remote inspection the risk is unacceptable the owner will negate his own insurance. If you advise your insurers they will be largely powerless to intervene. If anything I would be concerned (and it is a very remote possibility) that they might withdraw your insurance cover.
As a footnote I would add that he is not obliged to get expert inspection of his trees. Rather, recent case law suggests that he doesn't have to if he has had a quick look and sees nothing obviously wrong. However, that level of duty might increase if your expert or specialist raises specific concerns.
Based on what you have said so far, I would advise a specialist report, but make sure the specialist knows what you need and why and that his report will address it. I'm not touting for business, but if you are stuck I will inspect and report for you, as you are nearby. But generally I would not like to see Arbtalk being used in that way.
I hope my thoughts are of some help meantime.