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.
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.