Providing the tree is not protected by a tree preservation order or is within a conservation area then yes, you can do this legally if the branches are causing a nuisance. By rights, you should offer the arisings back to the neighbour as it is in their ownership. It becomes problematic if you must access the tree, e.g. by climbing, as this is then considered trespassing.
In our experience, it is always best to consult the neighbour first, or if you don’t get on, allow a third-party, e.g. a tree surgeon, to approach them on your behalf. This often results in the required outcome as a professional approach can reassure a neighbour that the work will be done correctly and therefore make them more forthcoming.