During the Waterloo battle scene of 'Jonathan Strange & Mr Norrel', a French attacker on the British line wades in with a battle axe. Did soldiers of the era sometimes carry traditional weapons for close quarters combat?

The battle axe you might be mentioning is a standard axe given to most engineers and sapers during the era. The purpose of the axe was to clear brush and other obstacles in the path of a battalion but as you mentioned it can be used in self defense. However it was only given to those men as others would have a musket and bayonet or a sword for self defense.