It was not just the Italian use of chemical agents like sulphur mustard and diphenylchloroarsine that escaped the eyes of prosecutors after the war, but also their use of concentration camps and killing of civilians in places like Yugoslavia and Greece. But you asked why these crimes were ignored. The answer is largely that the Allies viewed Pietro Badoglio specifically as being necessary for fending off communism in the new post-war Italian state. Subsequently, they did not want to alienate Italy. This is especially true for the UK, who did not favor an occupation of Italy like what would happen with Germany or Japan. Complicating things further was the fact that Britain had technically recognized Italy’s annexation of Ethiopia back in 1938, and such tribunals would shine a light on just how brutally that annexation was carried out. They knew Badoglio’s name would be at the top of Ethiopia’s list, and didn’t want him to be found guilty.

The most influential party here was the British Foreign Office, who during the war was led by future prime minister Anthony Eden who said it would be preferable for Italy to be “run for us, as far as possible” by Italians as opposed to direct management.

During the war itself, the desire of the Americans to bring the war criminals of Italy to justice was sidelined by the need to get Italy out of the war as soon as possible. This sentiment was best captured by FDR when he cabled Churchill to write that

We can secure the person of the ‘head devil’ [Mussolini] and his assistants in due time, and then determine their individual degrees of guilt for which the punishment should fit the crime.... The war criminal problem can be taken up later, and I believe that all demands by the Allied Nations that are not essential to the present time should be postponed with the purpose of getting Italy out of the war at the earliest possible date.

While the British foreign office wasn’t really expecting this change in tone from Roosevelt, they gladly accepted it. This was not the end of the questions, though, as the topic came up again when Anthony Eden was asked some questions in the House of Commons by MP Adams. It went a little something like this:

Adams: “Are you gonna bring Badoglio’s actions to the attention of the UN war criminal tribunal?”

Eden: “That was before World War 2. We’re gonna make an exception for China who wants to deal with Japanese war crimes before that, but whatevs. Oh also that’s not just for us to decide, but all the Allies.”

Adams: “Would not one of the purposes for which we’re fighting be frustrated if we let this sadistic sack of shit off the hook?”

Eden: Doesn’t respond

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After Badoglio pulled an Uno reverse card by signing an armistice with the Allies, people started noticing that Ethiopia had been left out of the UN War Crimes Commission (set up by the UK). As this was an odd exclusion, it was questioned in Parliament. Along with the excuse from earlier, the Foreign Office was like “uh they also stopped fighting in 1941 when their country got liberated” despite the fact that Ethiopia had offered to send troops to fight with the British (who refused the offer). This came up again, and the reply from the foreign office official (Richard Law) was “well they weren’t there at the beginning, so...” before he quickly realized it was a dumb response. He decided to wing it, saying that “I can assure the honorable member that the Ethiopian government were informed at the time these negotiations began and that they offered no comment on them”. This was a bold faced lie, but since nobody in Parliament knew the truth it wasn’t questioned.

International politics were soon to throw a wrench into the plans of the foreign office. At the Conference of the Italian Peace Treaty in Paris, the decisions were made by 2/3rds majority with the Communist bloc holding 1/3rd of the votes. By a happy coincidence, this made Ethiopia a swing voter that could make or break the interests of the West. Now in a position of power, Ethiopia was able to get this into the final treaty:

The date from which the provisions of the present Treaty shall become applicable as regards all measures and acts of any kind whatsoever entailing the responsibility of Italy or Italian nationals toward Ethiopia, shall be held to be October 3rd, 1935.

In effect, the treaty now said that the stuff about handing over war criminals Italy had agreed to applied to the war in Ethiopia too and not just what happened from 1937 or 1939 onward. The Foreign Office saw that its main argument had been neutered, and began insisting that it was now a matter between Italy and Ethiopia... two countries that did not have formal diplomatic relations.

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Ethiopia went to the UN, basically saying “we can compromise. We’ll only limit ourselves to charging 10 people with war crimes, and if Italy agrees to hand them over the majority of judges will be European”. At the top of the list were Badoglio and Graziani. The advocate general for the UN agreed to have their commission take a look. I will summarize what happened in the commission:

UK representative: “Hi guys! I’m glad you Ethiopians came prepared.”

US: “Can’t relate, this was on short notice so the government didn’t really tell me what to do here. Imma abstain if that’s cool with you guys.”

Committee starts reviewing cases, beginning with Badoglio’s

Ethiopia: “Here’s all the evidence showing the use of poison gas and bombing the Red Cross and all that fun stuff.”

UK: “This Badoglio guy has this plausible deniability thing going for him, it’s not like we know for sure that he shared responsibility, even though he was in command of the entire northern front. It was totally Mussolini, and uh... Graziani.”

Australia: “Bullshit.”

Norway: “Bullshit.”

Czechoslovakia: “Bullshit, crumpet boy. Badoglio was commander in chief and Graziani was only in charge of the South, there’s no way this could have happened without his knowledge or involvement.”

UK: “That seems like a you problem.”

Baron Leijonhufvud, the Swedish guy acting as Advocate-General: “What’s the issue here, UK? Badoglio even admitted the stuff about the poison gas in his book.”

UK: “Ok fine, maybe we can try him for the gas stuff but not the Red Cross bombings.”

Leijonhufvud: “This is first time in history the Red Cross got targeted systematically like this, there’s no way this wasn’t policy.”

UK: “Fine, fine, Ethiopia has a good case for both those things.”

———

By the end, the committee agreed that Ethiopia had a worthwhile case on its hands (Pietro and 7 others). The Foreign Office was predictably not pleased. Ethiopia also faced the issue that they technically didn’t have the authority to bring in the Italian war criminals for trial. Here’s how that went:

Ethiopia: “Help us, UK”

UK: “Ask Italy first”

Ethiopia: “Italy, give us your war criminals please. We’ll even compromise by dropping charges against everyone except Badoglio and Graziani” drops charges

Italy: “No”

Ethiopia: “UK, they said no”

UK: “Sucks to suck”

Italy: “Good news, guys! Graziani’s in jail for being a Nazi collaborator. Oh no wait, he’s out after a few months. Oh well. We cool, Ethiopia?”

Ethiopia: “... damn it. Hopefully if we drop this the Foreign Office will be more receptive to our claim over Eritrea.”

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TL;DR: The UK saw Badoglio as key to keeping Italy in the Western bloc, and didn’t want to alienate the Italian people by putting their leaders on trial for war crimes. The Foreign Office used its influence to keep Ethiopia from the UN war crimes commission until it couldn’t anymore, and then Ethiopia was shoved into the corner with Italy and told to talk it out which predictably didn’t work out.

Sources:

Italian Fascist War Crimes in Ethiopia: A History of Their Discussion, from the League of Nations to the United Nations (1936-1949) by Richard Pankhurst (1999)

The use of chemical weapons in the 1935-36 Italo-Ethiopian War, Lina Grip and John Hart (2009)