...we obviously realised that more people would be here in the summer but didn't realise that the children wouldn't be supervised and/or kept under some sort of control.
YABVU to expect that the parents would spend their holidays shushing their children near the pool.
Even at my local public pool (not in the ME but we get very hot summers) you can hear the din as you drive past, and this at family swim time with parents or caregivers present. You constantly hear the lifeguards shouting 'WALK' to the kids racing around the pool perimeter. They are kids. It is summer. The lifeguards don't take it personally that it goes in one ear and out the other.
British tend to have the kids be seen and not heard approach more than other cultures.
YY to this.
We've been in apartments the last 3-4 years so have had neighbours above, below, left and right (nightmare!).
There are a few more British assumptions and British conditioning at play here.
The kids can be in the pool constantly, but does that HAVE to mean that others can't? Does it HAVE to mean that others can't relax in their houses, gardens, wherever without the constant noise?
No, it doesn't have to mean that others can't relax or go to the pool. Others choose how to react. You could choose to go with the flow and check your anxiety levels. Mindfulness, acceptance, a sense of perspective would all go a long way to a happier you. Policing the pool behaviour, setting up rules that have to be policed, staying out of the water until there is nobody else there - all of that brings you back to your life in Britain. You are living somewhere else now, moreover somewhere that is hot and where there is a pool, and families with children.
I'm absolutely dreading getting kids on the same side as us
Maybe stop the dreading? You seem to be in the grip of an unusual amount of anxiety and pissed-offiness.
I have observed that there are some British types who do not feel completely comfortable unless there are rules posted everywhere and a good deal of enforcement. The urban area where I live draws a good few tourists and it is interesting to people-watch near one particular attraction where you can run around in a fountain that springs out of the pavement.
You can't seriously not see the issue here though?
Consideration, manners, respect for others. Are these all just old fashioned values now
This is the sort of tutting I have heard in various British accents - and only in British accents - while all around them there are adults and children of all nationalities getting soaked, yelling, children zipping in and out of the water and everyone having a good time, with the whole spectacle bothering nobody but the people who can't cope without the rules.
The common denominator in all this upset (neighbours in an apartment building and pool scenario) is the OP, who is not living in her native Britain any more and apparently finding other nationalities' norms hard to deal with.
(Agreeing with @TheHulksPurplePanties)
And you should install AC. I know it is against the rules, but take a walk on the wild side and do it anyway. The rule against AC is ridiculous and in any case, who is checking? You could possibly find an energy saving model to install in your top floor so the cold air can filter down. Run it at night.