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Hanging was the official method of execution in the British Isles until it was abolished in 1964.
Tolkien does state that certain offences are punished by death in Middle-earth.However, he gives us no indication of the method used.
For I am commanded to slay all whom I find in this land without the leave of the Lord of Gondor.
And the King said to Beregond: ‘Beregond, by your sword blood was spilled in the Hallows, where that is forbidden. Also you left your post without leave of Lord or of Captain. For these things, of old, death was the penalty. Now therefore I must pronounce your doom.