The British army did not have anything like this in terms of summary execution. Men would be arrested and court-martialled if they refused to take part in an attack, unless a medical reason was found.

Whilst some men were executed after a court martial for cowardice it was extremely rare. In fact of most cases where someone was found guilty their punishment was downgraded to military prison and dishonourable discharge.

In order to be executed the sentence had to be ok’d by the theatre commander, and by the time it reached them the case would have notes from every officer under him in the chain of command on what they felt should happen, from company/battalion level up to theatre command (French/Haig). Of the more than 10,000 cases that were put on their desks only 3/400 were ever executed and most of those were for murder or other violent crimes, not cowardice or mutiny etc.

Furthermore 0 Australian troops were ever executed despite carrying out a larger number of crimes compared to their contemporaries from around the Empire as their Governor General insisted on making the yes/no decision to execute any men from his nation, and he said no to each and every case ever put in front of him.

The book ‘Mud, Blood and Poppycock’ by Gordon Corrigan takes an excellent in depth look at this side of WW1’s history and shows that the British military at least, very rarely applied the death penalty and when they did it was for extreme cases where they felt an example absolutely had to be made.

You may be interested in this answer from /u/Georgy_K_Zhukov about the Red Army's "barrier troops", and this answer from u/klarok about deserters and mutineers in the western Allied armies during World War I.

I can really speak most about the French side of it. I will say they didn't specifically have "Trench Wardens" but some officers took the opportunity to execute soldiers for "cowardice" in the start of the war.

They used a system called the cours martiales, which were basically court martials. However, the severity of the conflict at the start for France made these a rather hasty process often abused by officers and led to speedy executions. The Military Code of Justice used in 1914 had been legislated it 1857, and saw little revision as the Dreyfus affair in the last decade of the 1800's made revision a sensitive matter. Military tribunals were to be done at the divisional levels for court martials, however during wartime officers had the right and duty to execute summary "military justice" by shooting fleeing soldiers. There is a great excerpt from Louis Barthas memoirs of the war Poilu in which an officer threatened to shoot him and a few others for not moving forward. However, when the officer was killed right after it was claimed by the men who recovered his body that he had not loaded his pistol, showing he maybe never had the intention to actually shoot the men.

Officers ability to get away with killing men for cowardice was basically down to lack of oversight from government officials. Due to the German threat on Paris in the first months of the war, the government had moved to Bordeaux and this distance allowed for Military leadership to basically run wild with little say from the national government. This included enforcing more severe court martials and executions. Soldiers were not allowed an appeal and the sentence was usually carried out the next day. This remained a trend until the government returned to Paris in 1915 and a proposed abolition of court martials on December 12th, 1915 and actual abolition in April of 1916. Even Pétain, the man known for his light handedness during the 1917 Mutinies, was not afraid to reduce what he saw as "mediocre" officers ranks and shooting "cowards".

One of the most egregious acts of harsh military justice for "cowardice" was what became known as the Souain corporals affair. General Réveilhac ordered the 21st Company of the 336th Infantry Regiment to retake positions captured by the Germans north of the village of Souain in the Marne. However, the artillery that was suppose to support them landed short and left the German lines virtually untouched. When the first wave went over the top, they were almost all gunned down immediately by the undamaged German machine guns. The remaining soldiers who were already exhausted by previous fighting refused to go over the top. To combat this, General Réveilhac ordered Colonel Raoul Berube, the commanding officer of the Divisions artillery, to drop shells onto the French position to force the men to move. Berube refused to execute the order without it written out by General Réveilhac, who never did. When the assault ended in a failure, General Réveilhac ordered that the men be punished. Company commander Captain Equilbey was ordered to provide a list of six corporals and eighteen enlisted men chosen from the youngest of the squads, with General Réveilhac court-martialing all 24 as an example. All the men were given a death sentence at their hearings. However, the eighteen enlisted men were pardoned due to them being arbitrarily chosen from the ranks. Two corporals were given clemency as they were never issued an attack order. Four corporals were still executed, with the French High Command commuting their sentences to force labor two hours after they had been executed. The men were finally exonerated in 1934, almost nineteen years after the had been killed.

This tendency to be heavy handed in the first part of the war can be shown with the number of executions that were carried out and comparing them to the 1917 Mutinies. The months of September through December saw an average of forty executions, with a peak in October of seventy-eight verdicts given and fifty-five carried out. A little over one hundred executions were carried out in the 1914, a period of roughly five months. Compare this to 1917 where there was only a third of those in 1914 at around thirty-three executions for the whole year, with the monthly average remaining around only seven executions a month for the years 1915 through 1918.

When it came to the mutinies, most of the action of the men was simply refusing to go over the top. There was no large scale violence towards officers, with very few accounts of it happening. Most of the action done was men getting drunk and then sitting down with their arms folded in front of them, which then became a sort of symbol of the movement. French leaders such as Pétain knew that just executing men was not the way to resolve these issues as it would do little to address the actual issues faced by the men, would destroy morale even more,and there was just too many mutineers for it to be anywhere viable. This is why so few men were actually executed in this period. Mostly ringleaders of uprisings were the ones executed, however even then many had their sentences commuted to prison time or were pardoned.

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So no, at least for the French Army there were not really people who had the specific duty to shoot people who didnt go over. It fell more to overzealous officers who took the opportunity from the lack of government oversight to enact how they thought the war should be conducted. There was no large scale order or protocol followed that resulted in a high number of executions comparable to Order No. 227 (Not A Step Back) carried out by the Soviet Union that saw hundreds of executions for "cowardice" and the creation of penal battalions on dangerous parts of the front.

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Sources:

-Barthas, Louis, and Edward M. Strauss. Poilu: the World War I Notebooks of Corporal Louis Barthas, Barrelmaker, 1914-1918. New Haven: Yale University Press, 2015.

- Greenhalgh, Elizabeth. The French Army and the First World War. Cambridge University Press, 2014.

-Watt, Richard. Dare Call It Treason: The True Story of the French Armies Mutinies of 1917. Dorset Press, 2001.