K, it's not what you want to hear, I know. But best this is out in the open and gotten out of the way. They have been stewing on this since they found out the house was for sale. And they probably didn't want to come and bother you while you were moving in. But now that they have your attention, they can bring up this point. I know from your veiwpoint it looked like bad timing...but for them it's been an issue for a long time.
I cannot say that neighbors are different in this country versus the US because I have never lived in such close proximity here, and had such a wide berth there, nor been in an area where there are so many semis that every house in an estate agent's window is assumed to be semi-detached unless otherwise stated it isn't. For that reason, I suspect neighbors in the UK have become much more territorial.
I know the neighbors we used to have were from hell. Trust me from my own experience: Get to know your neighbors, talk to them, compromise and let then know anything you want to do that may affect them (garden changes, home repairs, etc.). An ounce of prevention, ma'dear.
You think just moving meant freedom? Mwahahaha! Welcome to the neighborhood, kiddies!