For the yanks and elsewhere, yes conkers is well known in Britain. You basically put a chestnut (but its a conker) on a string by making a hole in the middle. Take turns swinging them on the string, whoever's breaks is the loser.
It used to be great fun till it was banned/requires eye protection now. There's an opportunity there, someone could make a perfectly safe conker app. I'm sure that would adequately replace it. /s
The reason I think this game is so popular is horse chestnut trees are very popular in the UK. For about a month each year, where I grew up the ground would be littered with conkers, both on my route to school and on school grounds. It's natural when walking around to try to find particularly large / impressive looking ones.
I do wonder if by banning it in schools it will get less and less common till it disappears. I suppose you predictably think that's nonsense.
But pedantry aside, its banned for the people who used to play it most and enjoy it, at the place they did just that.
Believe it or not the world has come on a long way in 16 years.
How could I be so clueless?
As for "the law" - from a 2019 petition to make conkers legal again:
There's no law or government policy banning children from playing conkers, so we're not sure exactly what you'd like the Government or Parliament to do.
“What exactly are you so happy about?' Harry asked her.'Oh Harry, don't you see?' Hermione breathed. 'If she could have done one thing to make absolutely sure that every single person in this school will read your interview, it was banning it!”
EDIT: And now I feel guilty for doing so.
The ban is generally lifted or just not enforced at all after a year or two.
For starters the playground monitors have more important stuff to do than remember the list of banned activities and try to enforce them.