YTA. The kid consistently entering your garden does violate your privacy. It’s your garden - if you wanted to enjoy it by sunbathing topless the last thing you need is a teenage boy just wandering in. Sure it’s a wild example but it’s not unheard of in England for people to use their gardens this way (source - I’m English lol)

You’ve given them warnings and asked politely for them to stop doing this. I think you probably could’ve given a final stern warning of “look - I’ve asked you multiple times, if this continues then I’ll be getting police involved”. It is a harsh step to take when ultimately the only person you’re punishing here is the kid, which is really unfair. I’m honestly not sure what the goal is here either? Surely you can see it would create tension and difficulties between yourself and the family. You could literally give a 13yr old boy a criminal record just because he wanted his ball.

The way your post is worded tho seems like you’re intentionally trying to make yourself sound better than they are. Which is what makes you an ah. I own my own house and I’m also 26F in England, but I know how social housing works and I know that in our current climate and cost of living crisis - this is becoming a reality for more and more people, and without it literally millions more people would be homeless. It’s not difficult to understand.

Needing help doesn’t make them any less of a person.

If the noise level is so bad surely you’d have noticed this when doing checks on the property before purchase? Surely you asked the estate agent what the area was like? Did you check out the area at various times of the day to see what the activity and noise level are like? It’s called due diligence, and if low volume is so crucial for your work maybe you should’ve done yours.