I really feel for you, OP. The fact you have been in contact with the council pointing out the missed bin collections is helpful - do you have a paper trail for that to show social services? You can't be held responsible for the failure of the council to collect.
I think you need to address this with your neighbour, though. If it was me, I would go round there. I wouldn't shout or be nasty, but I wouldn't leave until they were 100% clear about how ridiculous this action was, how serious a referral to social services can be, the communication with the school, and how much stress it is causing you. I would meticulously detail the huge list of other more sensible and empathetic alternatives they ignored: speaking to you about it; popping a note through your door; offering extra capacity in their own bin to help you out; if they were concerned, asking if you were OK or needed help; contacting a more appropriate department such as environmental services. Basically, I would make sure there was absolutely no doubt in their mind that they really f'd up here.
The extra bin to store waste out the back is a good idea while you wait for a solution. When we lived in London, we used a waste removal service a couple of times (clearabee I think - it wasn't cheap but worked well). If it's clean bags of dry waste, a mini cab might take be willing to take you to the tip. Not sure if you have a street WhatsApp, but people on ours often ask if anyone has spare bin capacity. It's insane that your neighbour didn't offer to let you put some bags in their bin if it was bothering them that much.
Some of the posts on here reveal a real lack of understanding of the fact that we don't all have the same time and access to resources. We don't all have a horse to feed our turnips to, a garden to compost in or a car to drive to the bottle bank. If we do our supermarket shop via public transport in the mad rush home from work to collect a child from after school club, then recycling anything marked "use local supermarkets" would mean transporting that stuff to work on the bus / tube. It's just not feasible when there are 38473289 other demands on your time.