Advice on Pear tree please

My suspicion is that your neighbour has a walnut tree (fast growing, large leaves, no blossom or fruit, etc); and what you have found are crown galls, which are nodules of bacteria similar to mycorrhizal fungi. Its time to cut a trench and sever any roots you find, say 1 foot inside your boundary.
Walnuts are capable of suppressing other trees from developing by secreting
juglone from their roots

 

Pollarding was the ancient practice of cutting a tree off at a level where livestock couldn't reach the regrowth; then after a few years the long growths were harvested and used. So similar to coppicing, but above foraging level often used to be done in deer parks and you can still see the evidence of pollarding on some old oak trees.
As for the galls as above sever on your side of the fence and I'd consider putting down something like the root barrier used for bamboos,

 

Thanks both, lots of digging ahead for me then, ooh me back

 

I wouldn't like to guess what the trees are from those pictures but they appear to be surface rooting, well at least one of them is.
I'd probably just cut out any roots that you find as they dont appear to be affecting your tree in any way.
Would be nice to work out what the galls actually are as they probably aren't suppressing the growth of the trees.

 

Sorry for the delay in replying, this is the best pic I can get of the cleaned up 'balls' they started to fall apart so had to stop brushing the mud off...

 

Well it does seem like it could be nitrogen fixing nodules if they were falling apart, but wouldn't like to say for certain.

 

Those thin black roots in your original photo could be the 'bootlaces' of Honey Fungus.
If you try and unearth some, check that there are no H-shaped junctions between them, where one bootlace joins another - normal roots don't do that. I'd also lift that bit of bark just above where they enter the trunk, to see how they attach to the tree.
Here's a photo of some bootlaces in my garden.
You can see the H-shaped junction in the middle.

 

Looks like Pear Root Galls to me