For your own sake and sanity, ivy CAN be a total nightmare - I know someone who put what they thought was a dead ivy house plant outside … only to spend the next 4 decades fighting an almost losing battle to stop it taking over the garden and house walls after it made a miraculous recovery! It’s very difficult to get rid of once it’s set in and takes a lot of maintenance to keep under control …
So lovely as it does admittedly look, if you have other priorities, it might be an idea to look now into how it could be more easily controlled or if you need to get rid of - not for your neighbours sake, but for yours.
Your neighbours rudeness towards you is completely out of order and you are right they have no idea what is going on in your lives. To play devils advocate on this however, conversely you also don’t know what’s going on in theirs. If they have previously been no trouble and suddenly are being particularly rude about this, could it be a final straw on top of things that were nothing to do with you? Or could they feel like they’ve said something to you and it was disregarded? Or maybe they just didn’t realise how they came across. Or maybe they did and saw the situation entirely wrongly (eg my mum thinks it’s lazy when I don’t put something back where it goes immediately, when in reality I’m trying to streamline the process and put everything away collectively as one quicker and easier - and less exhausting - task).
I do honestly believe good neighbours are worth their weight in gold and getting on with your neighbours is very important. And if they’ve managed to be nice people for 4 years, I hope for you that this is just a glitch that can be resolved. I do know that some people are just always going to be nightmare neighbours who want everyone to do everything the way they want things, and they often take a lot of compromise on one side just for a quiet life, which is often unfortunate (but when the other party ramps things up, so do they until the entire thing escalates and someone ends up on the wrong side of the law - honestly it happens!).
On the other hand, sometimes people bite their tongue about lots of little things to the extent they all build up and eventually the tiniest thing is the one to set it all off - the straw that broke the camel’s back so to speak - it’s the equivalent of why it seems sometimes like someone exploded and lost the plot for something as minor as, I don’t know, there being no paper in the printer, or the milk running out, or whatever. Maybe the ivy has been bothering them for years, or maybe they’re taking out on you the frustration of the weeds and overgrowth from the garden at the bottom that you mentioned.
If you would like to remain on good terms with your neighbours I would recommend having a nice calm chat with them. Either invite them over for a coffee/tea or just go around yourself to see them. Make it clear you’re not there to argue, but you would just like to resolve things and understand. Perhaps they’re now struggling to keep on top of their own garden and this added to their stress.
I am by no means saying bow down to whatever the neighbours want - they aren’t perfect in terms of their garden either! Bamboo is just as invasive if not more of a pest than ivy. It not only grows tall and blocks light, but if not carefully contained it has this nightmarish habit of putting out roots that spring up elsewhere as additional bamboo plants metres away from the original one! It’s possibly even more difficult to get rid of than ivy, and incredibly invasive. But as it’s illegal to mess with their plant other than what’s overhanging your garden, same as they can’t mess with your ivy, it may well be something you’re both going to have to reach a compromise on - if you are as fed up of the bamboo leaves overhanging. Just don’t turn it into an argument or point scoring issue.
I’ve witnessed too many neighbours falling out with each other over seemingly minor things, and people starting to allow pettiness and/or obsessions with what the neighbours are doing to take over. As I said before, good neighbours are invaluable - so, if you’ve previously got along with them I would suggest a quiet calm chat. Explain that you want to understand and resolve things amicably without an argument. Maybe it will help you understand what stresses they have going on in their lives and they to understand what’s going on in yours, they may not see the difficulties and challenges you have juggling work and family life - as well as maintaining the garden (including ivy and bamboo!).
Finally, in response to your comment about whether it’s down to their age and so many people’s responses that you’re being ageist and undermining your other concerns with this … it may well be age related albeit not in the sense of having nothing better to do (in contrast I know elderly people who maintain they haven’t got time for the minutiae anymore, either because of having so much now to do, or because things now take longer) - but perhaps previously they coped with maintaining their garden and the (possibly perceived) threat of invading ivy, and now they’re afraid ivy peaking through the fence will take over and soon they’ll have a garden like the overgrown one at the back). I mean they COULD just be cantankerous old folks … but they also might be really nice folks with their own as yet unknown reasons for being upset over 10 ivy leaves.
Please just have a chat - nicely. Perhaps take them some cakes or a bottle of wine/beer if they drink - say you want to try to resolve things with them - and hopefully you can restore the relationship and everyone can live amicably and with their own preferred environments.
Good luck - hope this all goes well!