I think a lot of the PP's on this thread are proving that they are not good neighbours or indeed, parents!
It is one thing if a child is having night terrors/meltdowns or something that is beyond the parent's control-but this isn't the case here. They're waking up at 4AM and having screaming and shouting arguments with each other.
This can be stopped by the parents, or the parents can remove them from their room and take them downstairs when this happens. I know-shocking that they'd be expected to actually parent their own children.
The parents could reinstall the soundproofing that they removed during renovations. They could switch the rooms so that it's not adjacent to the OP's and doesn't cause her disruption.
It's almost as if there is absolutely things that can be done to prevent their children causing massive disruption to their neighbours. Not just throw their hands up and say, "Oh, they're kids, what are you gonna do?"
It's just lazy as fuck. I'd hate to live beside any of the PP's who think that kids waking up at 4AM and having screaming arguments with each other counts as normal, family noise and that everyone around them should just have to endure it because, well, didn't you know that the parents have to?
OP- have a word with your neighbours. Don't do it at half 4 because chances are, anything you say will get ignored and you'll just get a pissy response. Go over during the day when they have a bit of time and just explain the situation to them. They may be unaware of the disturbance it's causing you. It's not always obvious how much neighbours can hear so they may not know it's waking you up on a regular basis.
Be polite, but firm. Mention that the issue started when they removed some of the soundproofing between your houses-that may give them a hint what they need to do.
And to the PP's who think this is just part and parcel of living beside people-no, it's not. Most people-the ones who are considerate of others-take efforts to limit their noise and disruption on others. There's noise that can't be helped-and that's part of the give and take. Then there's noise that can be helped but is not being helped.
No, the OP shouldn't have to move to some remote field just because she doesn't want to be woken up at 4AM every morning by the dulcet tones of screaming kids arguing with each other. Anyone who thinks this noise is reasonable and normal is a nightmare neighbour.