Aside from the United States joining the First World War, what other effects happened after the Zimmerman telegram was shown to the world?

Greetings! I shall attempt to give as best an answer as I to both sides of your question. As a (somewhat long) preamble however, it is necessary to note several things of importance that shall help set the context for these responses:

Firstly, the effect of the Zimmermann telegram on the US entry to World War I is often highly exaggerated by textbooks and simplifications of the war by media sources. The Zimmermann Telegram on its own hardly spurred any pro-war sentiments amongst the American public and government. What it was, in actuality, was the final straw on the camel's back. The American public was already growing increasingly upset by the German Empire's ambiguity with its war aims. What's more, it was growing clear to all the combatant nations that the anti-war protests and calls for peace in their own populaces were reaching a fever pitch. Woodrow Wilson in particular sought an opportunity to act as a mediator between the Entente Powers and German High Command (acting on behalf of the Central Powers). Safe to say, the exchange of peace terms which then ensued under his supervision went rather pear-shaped. Here's historian Michael Howard on the German debacle:

"The terms sought by the German High Command, on the other hand, were so extreme that Bethmann Hollweg (German Chancellor) dared not make them public for fear of their effect, not simply on the Americans [and Entente Powers] but also on the Reichstag"^(1)

Among other serious compromises on certain nations sovereignty (Belgium, Poland, and the Baltic territories among them), the demands included the condition that France not **only let the Germans keep Alsace-Lorraine, but give up the ore-bearing land around Briey.**^(2) The Allies, unsurprisingly, rejected these terms immediately.

After the failure of this attempt at peace, Wilson also had to deal with another troublesome issue: the continuation of German unrestricted submarine warfare. This had been called off in August of 1916, but it seemed an increasingly viable way for the German Navy to starve Britain as the war dragged on into its third year.^(3) With the failure of the peace negotiations, the High Command approved the resumption of "sink-on-sight" submarine warfare in the Atlantic and English Channel on January 9th, 1917. Critically however, it would not be until the 31st of January that the ambassador to Washington notified the American government of this decision.^(4)

Preamble over then, let's move along to the responses of the various nations involved. Surprisingly, the responses are fairly few and far between, those that fact is likely down to the simple fact that many governments and citizens were too busy fighting the war to worry about a telegram "across the pond". There are four main effects which shall be discussed forthwith:

1: Mexican Recognition: It is somewhat ironic that it took an invitation of war by Germany to get the United States to formally the government of Mexico under President Venustiano Carranza.^(5) This was taken mostly in an attempt to secure Mexican neutrality after the Zimmermann telegram, which was assured by Carranza. He could go no further in supporting the US though, as Germany was still a friend of Mexico and the US occupation of Veracruz in 1914 meant that supporting their northern neighbours would be the death of his popularity.

Carranza's Commission: Interestingly, upon receiving the telegram, Carranza immediately put together an official commission to look into the likely outcomes of joining with Germany in a war against America. The officials found overwhelming evidence that conducting such a war would never be in Mexico's benefit, convincing Carranza that the telegram was not worth pursuing further. Feel free to research the specifics of those findings, but here are a few key ones:^(6)

The promised German financial aid would likely not arrive, as Germany had failed to provide previously promised funds for the Mexican General Bank in 1916.
The American military was far superior to the Mexican one, and more importantly, it was united. Carranza's government was still weathering The Mexican Revolution, and declaring war on the US would likely incite even more civil conflict.
Even if, by some miracle, the US could be successfully invaded, Mexican authorities would find it extremely difficult to assimilate the southern states' English-speaking, armed, and independent populaces.

British Code-Breaking Victory: The Zimmermann telegram is also widely regarded amongst cryptography historians as one of the first and greatest victories of signal intelligence influencing global affairs. The efforts of Room 40 of the British Admiralty and its chief William Reginald Hall were applauded after the success of the telegram, though the series of events leading to its "leaking" to the American press is a most interesting read (and something quite like a spy novel).^(7)
Japanese Reaction: Crucially, the telegram also stated this bit:

You will inform the President of the above most secretly as soon as the outbreak of war with the United States of America is certain and add the suggestion that he should, on his own initiative, invite Japan to immediate adherence and at the same time mediate between Japan and ourselves.^(8)

The Germans had been trying to get Japan out of the war for a while, and Zimmermann hoped that Mexico could serve (if it did declare war) as a middleman between Berlin and Tokyo, mediating the peace by first making an alliance with Japan. The Japanese later issued a statement regarding their disinterest in joining Mexico or going agains the United States, and continued fighting on the side of the Entente until war's end.

Hopefully these brief vignettes of the few effects help answer some of your question, but I must also apologise as a historian for not being able to spend more time and access more resources (I have attached the names of several books below for further reading, should anyone reading this comment have the means to access them in efforts to satiate their own curiosity).

Sources:

1: Howard, M. E. (2007). The First World War: A Very Short Introduction. Oxford: Oxford University Press.

2: Howard, M. The First World War

3: Howard, M. The First World War

4: Howard, M. The First World War

5: Gilderhus, Mark T. "The United States and Carranza, 1917: The Question of De Jure Recognition." The Americas 29, no. 2 (1972): 214-31. Accessed December 18, 2020. doi:10.2307/979900.

Katz, Friedrich (1981). The Secret War in Mexico: Europe, the United States, and the Mexican Revolution.
Neidell, I. (2016, July 11). Mexico in WW1 - The Mexican Revolution I THE GREAT WAR Special. Retrieved December 18, 2020, from https://www.youtube.com/watch?v=vzsVUT7WqHA
Bellamy, J. (2016). The Zimmermann Telegram. Retrieved December 18, 2020, from https://www.archives.gov/publications/prologue/2016/winter/zimmermann-telegram