Generally, there should not be any responsibility for naturally occurring events like to the shedding and spreading of leaves in autumn.
I seem to be sweeping up several neighbours' leaves from our garden every November - and yesterday too !
Where there might be a problem is if this became a legal "nuisance".
The raw essence of nuisance is that an occupier's basic use and enjoyment of their property is being interfered with unreasonably by the actions of another party.
Nuisance is usually - but not exclusively - a continuing state of affairs.
The concept is a bit more sophisticated in practice.
If the tree in question is starting to create a scenario whereby the neighbouring property is being damaged by sprouting seeds incurring trouble and expense for the neighbour and effectively impairing the use and enjoyment of that property there may be a cause of action. That said the nuisance would need to be reasonably troublesome and not just trifling.
I would ask if the tree in question is actually suitable for the location. Generally, you can plant what you like but if the characteristics of the tree are such that regular impairments of neighbouring property are likely you cannot just ignore that issue. I can think of many who would ignore it and who are just bolshie neighbours.
What about the owner's property ? Are they not also having similar problems ?
I doubt that anything can be done to abate the problem other than removal of the tree. I would agree with your idea of consulting an expert on trees but I would think more specifically of an arborist/tree surgeon. Beware that there are "tree surgeons" and then there are real tree surgeons
Until you get advice from a tree surgeon and are able to make an informed decision before your next move I would not offer to do anything just yet.