Technically you are supposed to offer anything cut from a neighbours plant back to them so you would be within your rights.
Am I right in guessing that your council is now charging to take away garden waste and that's why you want to now give them the cuttings. if so I would simply politely explain your reasons for no longer wanting to dispose of them yourself.
But re neighbourly relations, would only ask whether this means you won't need to use the council service or will only Ned maybe say bin rather than two? Only because if you do have space in your bins then it would seem a little churlish not to dispose of the cuttings in yours and only give the excess to your neighbours to deal with.
its not the same issue re overhanging plants but my neighbour and I often 'use' each other's paid for garden waste bins when our own our filled (asked and arranged in advance of course) that way we can manage with fewer between us, saving us both money and space.