I feel sympathetic to you, Marybel. We had a lovely silver birch growing on one side of our garden. The neighbour adjacent to that side said its leaves were getting everywhere (he has one of those paddling pool 'swimming pool' type things) and also that the roots would start to interfere with his garden. We said we would prune it back so it wasn't overhanging his property at all, but we went out for a cycle ride one day and came back to find he'd leaned over our garden fence and hacked it back well beyond our boundary line. He also did the same to some silver birches growing in another neighbour's garden at the end of his.
We were pretty fed up about this (and he took the cut branches away, not asking us if we wanted them)... Then lo and behold, later he proceeded to dig down into the soil next to the fence and attacked the tree's roots. Within two days of the digging the tree's leaves all yellowed and dropped off simultaneously. I'm not sure if it was purely physical damage (root removal) or if weedkiller was involved, but the end result was the same: one dead tree.
This particular neighbour is the kind of bloke with 'anger management issues' and we didn't want ongoing conflict, so we didn't pursue it any further than just letting him know we were upset that he'd done this. As the birch had a bird nest box on it we decided to reduce its height but leave it standing as a dead wood feature: four years on it's still there (albeit getting a tad wobbly). I planted a rowan in front of it this year, which seems to be doing well. The neighbour has been less on the rampage of late (due to health problems) so fingers crossed the rowan will get a chance to flourish. If he does start waving a saw around I may just sneak into his garden at midnight and poke a hole in the side of his pool...

Or train the birds to go and poop into it at regular intervals!

Anyway, I can grow as many trees as I like from seed, so he's fighting a losing battle.
I hesitate to wade into the whole "Sycamores: Good or Evil???" debate as I've seen it aired on WAB several times, but just to add my two pennorth:
Pros:
Whether you consider them native or not (and the debate about this, based on the pollen record, could run indefinitely) they do support lots of native species.
Because they harbour high numbers of invertebrates, including aphids, they make valuable feeding grounds for insectivorous birds.
They are hardy trees which can tolerate exposure, wind and pollution, and establish quickly to make good shelter.
You can pollard or coppice them to control shading etc: this will also prevent or reduce seed production.
Sycamore timber is very good for wood turning (especially bowls and plates), spoon carving etc; it is also used for musical instruments. Just 'cos it's not the number 1 firewood of choice doesn't make it a worthless wood!
Cons:
Mature trees cast a heavy shade and will seed freely in good conditions. This can be addressed by pollarding or coppicing (see above).
Sycamore leaf litter can be dodgy on paved and hard surfaces (slippery when wet).
Left to their own devices they will become large trees, not great for a suburban garden with limited space.
I guess you could pollard it properly and see how things go with your neighbour... Or lop the top off but keep enough of the trunk to tie your washing line to (and maybe put up a bird nest box on?), then plant another species of tree nearby to replace it. Our native
Acer species,
Acer campestre aka Field maple is an attractive tree with lovely yellow autumn leaf colour. Or if you'd rather go for something smallish, there are several possible options: how about a crab apple, rowan, hazel, whitebeam or hawthorn? Or going smaller still, shrubs such as guelder rose and dogwood have lovely colour and flowers/berries for the wildlife.
Anyway, good luck with your tree... And your neighbour!