I’d guess from what you say that the land behind your place is common property. An adjacent land owner has a right to drain water over that lane. This right is called an easement. The easement does not convey any other rights; for example, the adjacent owner can’t build a shed on your land.
If there’s a tree on the land that interferes with the drainage, the adjacent owner can insist that steps be taken to keep the drainage working. Perhaps that requires the tree to be removed. However, your local council may have made a tree preservation order and you might need permission to remove the tree.
I’d also guess that any work required on the tree will be at your OC’s expense. But it would be nice for the adjacent owner to allow temporary access via the land next door if that would help.