Its sad, but I dont think there is any law that says they have to. I never let my garden overgrow into my neighbours properties. However this recently backfired when I moved into a new house a couple of weeks ago, trimmed back an enormous rampant rose bush which was reaching my neighbours balcony (but well inside my garden boundaries!) only to later overhear her moaning to someone on the phone that the new neighbours had cut down the rosebush she had spent years growing up to her balcony! Cant win!
I would tell your neighbours of the problems the tree is causing you and see if you can come to an arrangement. If not, cut it back to your boundary, even if this makes it unstable. You have every right to do that. The only other thing I could suggest is maybe buidling a covered carport over your driveway. It would at least keep your car clean (and the kids!)
My mums neighbours have got huge plum trees that grow halfway across my mums garden, they drop fruit all over mums garden, which in turn makes the garden a haven for wasps and the kids cant play out there! Tried to speak to the neighbours but they are polish so didnt really understand me, but they did cut back the trees-on their side, not touching the bits that have grown over mums
I moved to a house once with an elder tree very close to our neighbours garden, and she popped around very soon afterwards to ask us to cut it down as the birds ate the berries, crapped all over her conservatory and stained it red. We were happy to do this, surely most people would be OK with this?
Have you tried speaking to them to discuss the problem, don't they have the same issue over their driveway too?
I have spoken to the in the past and they have just said they can't afford it, the birds do poo on their cars too, but not as much, due to the angle of the houses, a car port would not only look VILE, would need planning permission and why should we pay for that?!?
I have offered to contribute, but again nothing, will have another go at talking to them.
Now their kids are mobile (baby and toddler last time) it might be more of an issue.
We had similar problem, though not as bad as yours sounds. We have a shared front garden, the lawn is shared but there was a very large sycamore tree on the neighbours half, which is 'her' tree.
It got very big, was starting to affect the light to our lounge and because our parking space was nearest to the tree, the car would get all this sticky sap stuff like a film, all over it at certain times of the year. As well as bird poo, and the branches were starting to reach into the actual car parking space.
We get on ok with neighbour, so we did pluck up courage to ask whether there was a chance of her cutting it back. She didn't mind, and actually said she wanted it cut down anyway, but couldn't afford it. So we had to put up with it till she got the money, as we couldn't afford to contribute....We'd just had to have the Leilandii's cut down on our side that previous inconsiderate house owner had planted.
Try talking to your neighbour if you get on with them. They may not realise what awful problems it's causing you. But I don't think there's much you can do if they decide niot to cut it down or back.