Germany was motivated by pragmatic as well as ideological considerations. Spain was an important source of natural resources like Tungsten for the German war machine, as well as a potential captive market for German manufactured goods. The bilateral German trade agreements in Southeastern Europe that secured minerals and agricultural goods without expending foreign exchange needed for rearmament were a model for German-Spanish relations. WW2 freed Spain from its obligations to German creditors, but otherwise it would have been part of an informal German economic empire that also included much of Southeastern Europe.
The influence of Hjalmar Schacht was important in this policy area, keeping the "economic fuhrer" out of power might prevent German intervention in Spain. Schacht was the Reichsbank President that ended hyperinflation in the '20s, the first nazi economics minister and financial mastermind behind nazi rearmament, and one of three Nuremberg defendants who was acquitted.
The lack of western support hurt the republicans OTL, but the lack of fascist intervention could make France less hesitant to intervene. Leon Blum initially considered fulfilling the republican request for aid, but soon backed off after a perfect storm of several factors. The failed coup that began the SCW occurred on July 17-18th, 1936, but the Spanish Ambassador to France was a conservative with rebel sympathies who dragged his feet on relying information from the government in Madrid to Leon Blum's government. A member of Blum's cabinet leaked the plans to aid the republicans to the German Ambassador to France, who created a diplomatic uproar that led Blum to back down.
In a scenario without foreign intervention or just a different course of action by the Spanish government, the odds definitely favored the Republicans. The republican government knew a coup attempt was coming about a week beforehand, but planned to let it go forward on the assumption that it would just fizzle out. General Sanjurjo was at the center of a failed coup plot in 1932, which is why so many nationalist military leaders had been sent to Morocco, and the government had been reshuffling and dismissing officers.
The plot began in Morocco on July 17th, but poor communication meant that coup plotters in northern Spain didn't receive the signal to revolt until the 21st, when they lost any element of surprise.
The Navy was on the government side, as was much of the air force (173 republican planes vs. 40 nationalist planes at the conflict's start), and the Army of Africa was stranded in Morocco. A timeline without intervention would see the nationalists quickly defeated, followed by tension between leftist militias and the more moderate republican government. The republican government opened its armories to communist and anarchist militias in order to defeat the nationalists, but these factions' loyalty to Manuel Azaña Diaz's elected government was questionable at best.