Although not quite the same problem I'm having issues in my first property I've bought.
I moved from a rented property in the city to get away from the loud parties and general noise of the city. I bought a 1900 terrace and naively presumed the walls would be thicker. Big mistake (now learnt) as the walls are like paper and I can hear everything. Even worse is that both sides are occupied by young families which both had 2 year olds when I moved in, now they both have had another child.
One side I get along with great as they are very respectful and try their best to keep noise to the minimum. Having a 6 year old myself I understand the frustration of tantrums and controlling a child. The key to this is respect which the other side seems to lack. The child is like a demon and has regular tantrums which I know is hard to control and remain calm with. The problem is with how they talk to the child, scream at it, do not give it regular bed times (one night its 7pm then its 10:30pm when they get in) also staying in bed till noon and shouting at your child to go back to bed all exacerbates this problem. They seem to be on the uneducated side and always talk loudley in the same dreary voice. Also shouting from downstairs to upstairs shows a lack of respect to neighbors both sides.
Unfortunately I can hear all this and can't help but become accustomed to listening and knowing their habits. For me its a major let down and partly my own fault but on 3 viewings the property was quiet so what else could I have done. I also have anxiety issues with sensitivity to noise so yeh I'm pretty ****ed at the situation.
It's been 14 months now, the property needed no work and is in a good sought after area so not long until I can look to jumping somewhere else.
Personally I get frustrated that we have to listen to other peoples lives, especially those that show a lack of respect which is key to me and my level of stress. The only safe option is a detached eventually which I hopefully will get to. I sometimes feel extremely unlucky but I know there is worse and it goes on everywhere. For now filling the house with radio, tv and ambient noise apps on my phone has really helped especially with sleep while screaming babies are going off.
Good to get this off my chest even though offering no advice, its crazy that living with such noise is just accepted as part of life (tolerating the noise of other people's lives, perfect world...)