conker

That is a conker tree.

Have you seen Tom Smith's conker?

when one conker has broken another it becomes a oner.

The medicinal qualities of their conkers, leaves and bark soothe varicose veins.

Horse chestnut The medicinal qualities of their conkers, leaves and bark soothe varicose veins.

Somehow, miraculously, my conker-sized kiddywink brain managed to differentiate it from a real egg.

Well, I do have some skill at conkers, if you must know but I fail to see why that's relevant.