There is an interesting example of British military justice, though it is from 1775: When the British troops under Howe landed on the Charlestown Peninsula across from Boston before the battle of Bunker Hill, five British lower enlisted quit their ranks and attempted to desert to the American lines up the hill. They were caught and were brought to Howe, who singled out two of them and said, 'hang those two' which was promptly done. No trial, nothing but summary execution, which in this case was quite merciful, saving the other three.
I feel I should point out that the anecdote you cite is essentially fiction, presumably drawing on Thomas Robinson's 1960's semi-fictional work, 'Now we are Enemies'.
The only contemporary reference to British deserters being hanged before Bunker Hill comes from an anonymous letter supposedly from a British officer, which was quoted in the American press a remarkably short time after the battle. It appears to be a fairly transparent piece of rebel propaganda. General Howe is not mentioned, nor any discussion regarding the men to be executed.