For not wanting neighbours kids in my property?

Future burglar. 12 is too old for him not to know better.

Its a little odd the way you say 'my property' like they are commiting some huge crime or criminal damage.
Its a bit weird you told the child to go away then felt the need to go lecture the mother too. Surely just telling the child they arnt allowed would have been enough

The mum sounds VERY weird and entitled. Her child most certainly cannot do whatever he/she pleases when it involves trespassing on someone else’s property. Her child was not “just having a bit of fun”. Her child was either being a bit stupid (in which case she should educate her child about recognising garden bounderies) or being ignorant (in which case she should make sure her child looks where he’s walking).
Is this child nearly the size of an adult by any chance? In that case I would call the police next time it happens, reporting suspicious behaviour ( possible burglary?) on your property. It is not unrealistic to call the police if one sees suspicious behaviour, and if an adult was lurking around on people’s property a lot of people would think burglars.
And ignore the mum. She sounds like a total piece of work. No wonder the kid is wondering around. Sounds like he/she isn’t being taught basic manners.

This shouldn't be happening, for the child's safety too. How does the mother know you wouldn't do the child harm? I'm not suggesting you would, OP, but some would as we all know.

Can you lock your side gate?

Since you ask, no I would not be ok with a random coming into my house or garden without asking. Of course not, who on earth would be. Hence my gate is locked, as is my door!

I don't imagine you're the only person who lives in a property at the end of a road/cul-de-sac.

Child is too old to be doing that and the mother's reaction is appalling. A neighbour's child would try to follow me home and into my back garden once or twice and another child came with her friend (who was, at least, the daughter of a friend of mine) and let herself into one of the animal pens) but there was never a repeat.

Were they collecting a ball, or toy?

I suggest you wander into the mum's garden and start having a picnic. And tell her "Lighten up, it's just a bit of fun".

She's a CF, of course her 12 year old shouldn't be wandering into other people's gardens.

I live in a detached house in the bottom corner of a cul de sac, where my driveway leads down from the road and as a result no one passes my house at all, ever. There’s no path or road past my house. A neighbour’s child and her friend took to using my driveway to play on their scooters. It drove the dog round the bend and, as a result, drove me round the bend. I asked them to stop playing on my driveway and felt no guilt in doing so. YANBU

We recently moved house and one of my stepchildren is 12, has autism, and has twice now walked into a neighbours house thinking it was ours. She realised after opening the front door and stepping inside, stepping out shutting the door and running back to the adult she was with but both times we've been apologetic to the home owners but she's been quicker than we could get her. Our new street all the front doors/front of house look the same, only been 1 or 2 houses away from the right one.
Could it be an accident/unintentional or a neurodivergence?
I did apologise but I know not all parents would.

They are aware but kids that age sometimes do dumb things without thinking though the consequences or how it might make others feel. My son is generally well behaved and considerate but i would be open if someone came to me to point out he’d been up to something he shouldn’t have been doing. I want him to behave well for his sake and for the sake of others.

FGS, unless he’s got SN a 13 year old should be very well aware that unless they’ve been invited in, other people’s gardens are off limits.

Nope, unknown kids don’t get to just wander into your property. Do you animals she was trying to access? I’d tell her very firmly to never do that again.

When we were that sort of age exploring the neighbourhood did sometimes include checking out gardens, seeing how far/where we could get over back gardens in particular. We knew we shouldn't be doing it though and our parents would have punished us if they'd found out.

So I do find it quite "normal" but like most testing of barriers that children try, that doesn't make it okay or something parents should be okay about.

Not sure what you do about it if the parent isn't going to take action, though. Maybe have a hose ready to water the garden? Trespass isn't illegal, but if you have neighbourhood policing the local Bobby might have words as it's pretty anti-social.

Put a lock on the gate so no one can access your property.

Definitely weird! I would really not be happy at all

No, I would not be happy about it.

No, you don't just wondering in to someone else's garden through a sidegate of a property where you dont know the owners.

Its weird! And 12 years old is definitely old enough to know better.

The mum of the wondering child should parent her child properly.

What if said child hurts themselves on your property and you have no idea they are even in the garden?

The only people of any age I’d want in my house or garden are those I’ve invited or that are actual friends of my own children and invited by them. I might turn a blind eye to a child I knew who had just come to retrieve a ball over the fence. Otherwise I’d assume they were up to mischief and tell them to go away. I probably wouldn’t bother the parent though unless it happened again.

Age 12 is well old enough to understand basic social norms around not opening gates and trespassing into gardens uninvited and to realise some people would be upset, scared or annoyed by it. It’s also old enough that, rightly or wrongly, it’ll be assumed they’re up to no good.