Let's even say that there *is* a successful right-wing coup against Cardenas (which I think very unlikely). Why in the world would the regime established by such a coup want to join the Axis when this will obviously bring about swift military action from the US, and the Axis powers can obviously do nothing to help Mexico? The right-wing authoritarian regimes in the Caribbean area--Trujillo, Somoza, Batista (though by the 1940's he was less authoritarian and right-wing than he used to be)--were all strongly *pro-*US during World War II. And even the most Axis-friendly regimes in Latin America, like Peron's Argentina (which could of course afford to be relatively independent of US foreign policy only because it was far away) never considered doing any such thing.
Before anyone beings up the Cedillo rebellion, I'll just point out that (a) it had very little chance of success, and (b) it was *not* fascist nor was it backed by the fascist powers of Europe:
(1) "Owing to his ideological conflict with Cardenas, Cedillo was approached by many right-wing groups, including the fascist organization known as the 'gold shirts.' However, such links have been exaggerated by official and left-wing versions of the Cedillo rebellion, who have tried to portray him as an ally of the foreign oil companies whose Mexican holdings were expropriated during the zenith of Cardenas's Revolutionary nationalist measures in March 1938. While the oil companies undoubtedly were pleased by any efforts to overthrow Cardenas, the aging, increasingly ill, and militarily weak Cedillo never enjoyed any concrete support from Washington, the oil companies, or even the fascist governments in Europe, who saw little to be gained from such an adventure. Cedillo's rebellion was virtually stillborn. The federal government had made certain that it possessed all the necessary elements to defeat the caudillo from Palomas before forcing Cedillo to rebel by demanding that he leave his base in the Valle del Maiz. When he was forced to take up arms in 1938, it was basically a suicidal gesture in defense of his personal honor and a refusal to recognize that his days as the caudillo of San Luis Potosi and his local project in the Valle del Mali were a thing of the past. " *Concise Encyclopedia of Mexico* (article "Cedillo, Saturnino")
https://books.google.com/books?id=VeI4CQAAQBAJ&pg=PA96
(2) "At that time, German government representatives were hoping for a trade expansion with the German state, not a civil war that might only benefit U.S. economic interests. Later, after Cardenas's destruction of the Cedillo movement, the German Foreign Ministry conducted an internal investigation to determine whether any German ministry had supported the warlord from San Luis Potosi. The investigation failed to identify any German involvement. Moreover, the Cardenas administration itself assured the German legation that continued public accusations of German involvement in the Cedillo uprising were really part of a leftist anti-fascist propaganda campaign. Mexican Undersecretary of Foreign Affairs Ramon Beteta apologized to German Minister Rudt, stating "that the isolated smear campaign against Germany is in no way representative of the position of the Mexican government and people."
"Repeatedly, the existing literature has identified Cedillo's German-born chief of staff, Colonel von Merck, as a likely conduit for German influence. The sources in German and Mexican archives, however, show that von Merck established contact with Berlin only *after* the failed rebellion. Ina letter written by Colonel von Merck himself to a supporter in the United States, Cedillo's chief of staff displays the very domestic conservative nature of the Cedilla rebellion. Von Merk wrote to a friend in San Antonio:
"'what we want is that communism in Mexico, which the government unconsciously supports and which is gaining importance in Mexico, ceases to exist. I believe in nothing absolute, neither the Nazis not the Fascists who want to bring their doctrines to Mexico, which are perhaps good for Germany and Italy, but not for Mexico. We don't need to import anything to Mexico, not from Russia, Germany or from Italy, not from any other nations; we want a national nationalism, a Mexican one, nothing exotic. The politics of Europe is of no interest to me, only Mexico.'
"Once again, the strong domestic orientation of Cedillismo was confirmed. In addition, the U.S. and British governments refused to support Cedilla in any form. In April 1938, the U.S. embassy refused to see on Merck, in order to avoid any appearance of complicity. In September of 1938, Cedillo's sister tried to establish contact with the British Ambassador in Washington 'to give him the true facts regarding the situation in the state of Sari Luis Potosi.' She too was rejected..." Friedrich E. Schuler, *Mexico Between Hitler and Roosevelt,* p. 135.
https://books.google.com/books?id=q4ZiwTZpWoYC&pg=PA135
Of course, even if Cedillo *did* come to power (which is extremely unlikely), even if he had been backed by the fascist governments of Europe (which he wasn't) and even if he had lived past 1939 (which he didn't), he wouldn't be crazy enough to join the Axis or go to war with the US. But since it was claimed earlier in this thread that a right-wing coup against Cardenas was possible and would have been welcomed by FDR, I just wanted to give some evidence that such a coup was very unlikely, and that in fact FDR gave no support to the closest thing (which wasn't very close) to a plausibly successful attempt at one.