Head, mouth and bed are all river words, which is why it fits so neatly.
Edith Ordway’s 1914 handbook has a shorter form: what has a mouth but never speaks, and a bed but never sleeps in it?
What runs but never walks, has a mouth but never talks, has a head but never weeps and has a bed but never sleeps?
Give me a hintThe hint
Follow it and you’ll reach the sea.
Reveal the answerHide the answer
A river
Source: English nursery riddle; a shorter form is in Edith B. Ordway, The Handbook of Conundrums, 1914