Don't bother messing around with clerk or even the PC. What's the point? If, as you say, they don't own the land then they have no liability for tree(s) thereon.
If (if) you can establish ownership of the land, get a suitably qualified arborist / arb consultant to compile a tree risk assessment (you'll have to pay for that of course) and forward it by recorded delivery to the land owner.
The land owner is under absolutely no obligation to act upon the tree risk assessment but you will have a documented audit trail which may be relied upon in a future civil action for damages IF there is damage or harm caused by the neighbouring land owners inaction in relation to a known issue with the tree(s.)
The tree risk assessment negates any mitigation claim by the land owner that they had taken reasonable steps to prevent harm to others and the recorded delivery negates a claim that they were unaware of the potential hazard - this all comes under the Occupiers' Liability Act ('57 & '84.)
You may feel aggrieved at having to foot the expense of a tree risk assessment but the benefits may not be immediately apparent.
You state that 1 tree has already fallen and that some of the others are "dangerous."
There is a huge potential for those within the sphere of influence of tree(s) to over estimate the ACTUAL risk of harm from falling tree(s) or branches. This is where the engagement of a Professional Tree Inspector and/or a Quantified Tree Risk Assessor will provide quantifiable assessment of risk based upon Visual Tree Assessment, knowledge of tree species, age class, physiology etc. In many cases, such a report actually provides the reassurance that the tree(s) DON'T present a quantifiable risk and the person with the concern can get on with their life.
There also exists the inalienable right of 'self-help' to abate a nuisance which is being visited upon you. A person may remove limbs / branches / roots of tree(s) which cross a boundary. This right even exists where the tree(s) may be subject to Tree Preservation Order - especially if they create a statutory nuisance.
There are certain caveats attached to those rights to self abate a nuisance and the advice of a professional arborist / arb consultant should be sought to ensure actions are lawful. In essence, you must take the course of least mischief and informing the landowner in advance is one of those reasonable courses of action.
Don't bother with the PC - there's nothing they can do other than pass it up to LA. You could try submitting a "Dangerous Tree Report" to the LA but that will likely require you to have first identified the land owner anyway.
LA can (if they want to - it is a "discretionary" power) take action under s23 Local Government (Miscellaneous Provisions) Act 1976 to compel the land owner to remove a dangerous tree. Normally by serving notice and, if not completed within notice period, undertaking the work them self and placing a charge against the landowner.
Get some professional advice from a qualified arborist and forget wasting your time with the clerk / PC.
PS - my webpage https://www.empiretrees.co.uk - feel free to get in touch if you want more detailed / specific advice (pro bono)