Next door had their front garden sorted last year and it involved cutting down some trees that were in it. I've no idea why but they didn't bother having the dead tree in their back garden cut down at the same time.
With the storms earlier this year the dead tree popped one of our fence panels out and it now clearly leans across our boundary. We haven't been able to replace the fence panel as it's impossible to slide one into the supports because of the tree. Dh spoke to the neighbours about it (a few weeks ago iirc) and they said they'd sort it sometime 
as they're planning on replacing their fencing and will do it all at once.
However with the nice weather today we've been cutting our grass and I think the tree is leaning over even more than it was. It's windy today and the tree is visibly rocking quite a lot. I'm also concerned that it's uprooting as they've put piles of bricks/rubble/crap over the base of it as if to weight it down a bit . If it falls it's going to come down into our garden.
Have attached pics.
Is there anything we can do to get them to hurry it up? Tbh we don't really get on with them after a disagreement about the joint brick shed roof (they used a dodgy mate to fix their half and now we can't get a builder to go near our half with a bargepole). They tend to be a bit 'aggressive' iykwim.