If the coup of July 1936 had been an overwhelming success in Spain had been proclaimed a military junta headed by General Sanjurjo. However, details such as the form of government had been given the support they had had the coup. For example, the Carlist were extremely important in Navarra, but in the rest of the country were a minority force, and although the CEDA had many pro-monarchist elements, they accepted a Republic of conservative authoritarian.
There is also the question of whether Sanjurjo died when he was traveling to Spain (as in OTL), which would lead to a restructuring of the coup commanders, and if the coup had been an overwhelming success, their leader would surely General Mola (the true organizer of the coup), in favor of maintaining a military dictatorship under the republican form of government (which led to many confrontations with the Carlist in preparations for the coup).
However, I honestly think that in the event that the coup had succeeded it would have established a military dictatorship under the guise republican would have eliminated the major leftist leaders, based on the former dictatorship of General Primo de Rivera, but without a true political control of the Falange, establishing a similar group of families ideological (monarchical, traditional, military, Falange, Catholics, agrarians) that converge in a conservative military dictatorship, similar to the Franco dictatorship in OTL.
Surely, after the growth of fascist movements in Europe, the dictatorship would be likened some elements of the Falange to accommodate the new European situation (as did, for example, Milos Horthy in Hungary), but only Spain would enter into the Axis if Germany assured him Gibraltar and part of the booty French (French Morocco, the Oran region, Mauritania, Roussillon and an expansion of Spanish Guinea, without forgetting the possibility of conquering Portugal, with what it all meant, including the Portuguese colonies) when France was about to be defeated (similar to what Mussolini did in June 1940, but not before the Franco-German armistice).
Its further evolution would depend on whether, with the Spanish contribution (which would be a little less important than the Italian, but it was considerable), Germany could force Britain peace (only I could get if German could conquer the Suez Canal and joined Iraqi nationalist forces that fought against British colonialism) and then deal directly with the Soviet Union in search of its Lebensraum.
Otherwise surely be repeated a similar scenario to the Peninsular War, being able to produce a landing in Spain (always after another landing in North Africa, on French colonies, perhaps in Senegal), causing the Second World War ended not before 1946.
With this scenario, we can not deny the possible influence that the military dictatorship established in Spain in Latin America could produce, as it could fly the flag hispanist against U.S. imperialism, creating many socioeconomic conflicts with the U.S.