In North America where I grew up, the chestnut-tree thing happened,
only the tree was called a "buck-eye" tree
because of the 'conker's resemblance to the eye of a deer.
And the 'conkers' themselves were in fact called "buckeyes"
and we used to do the same thing Senoufou describes,
making a whole string of pierced buckeyes.
Goodness, that does bring back memories.
And yes, about this time of year,
the very old and tall buckeye tree in the front yard
would be shedding buckeyes all over the driveway,
where the passing cars would crush them . . .