Sorry for the missed questions.
I too thought initially that it was odd that the hole was on our 'side' (and so high up too). But we are on a bus route with regular traffic and a bus stop opposite, so it would have looked less suspicious to be working from our side. And it's easy to see when we're out if you spend as much time looking out of the window as the neighbour does...while this might sound paranoid of us, for example I did once come back to find his gardener leaning over the back garden fence and cutting two parallel horizontal lines through the ivy on our ash tree, about 18 inches apart, in order to remove that section and therefore kill the ivy above. He said he hated the ivy and that it was "smothering" the tree (not sure why he was bothered about that) and he wasn't at all embarrassed about being 'caught'. So you see, he has form. No effect on the ivy, by the way, as he didn't get to complete the job. And he hasn't tried since.
MattyF, are you saying this does look like a drill hole, but that you wouldn't have expected it to have been able to kill the tree?
Cornish wood burner, you have a point, especially as nothing can be done, and he'll certainly deny it. I'll think about that (but definitely mention it in some non-accusatory and puzzled way in passing the next time he wanders by with a complaint).
Billhook, I was thinking too that the hole might be too shallow for a chemical, but from what others have indicated it does seem possible. I then wondered whether something injected into the tree and therefore down to the roots could somehow spread to nearby plants? The weeds grow back there, so it's not as if the section of earth as a whole is completely resistant to anything living. Haven't tried to plant anything yet though to test that. I appreciate your recommendation about a camera but feel a bit reluctant to take that route as I don't want to become too (even more?) obsessed about what next door gets up to ie becoming like the neighbour himself.
Sloth, yes, the chuck-scored outer circle is exactly what it looks like. And thank you, I will see if a friend of a friend (there must be one) can have a look.
The tree is not very old (the section of earth was only put there 8 years ago, and as I say the tree just started growing there over time) and not very tall. Here are a couple more photos, including one showing the hole from above. I hope they're clear enough. The hole on the right hand part of the tree, about halfway between those two parallel branches.
Very many thanks again for reading this and for all your help.