I wrote a
timeline of a Republican victory almost three years ago. The PoD there is an earlier breakthrough in the Battle of Brunete stifles the Nationalist actions against Madrid and cuts off the path to the Cantabrian Mountains, leading to a drawn-out campaign in the North and bitter rivalries in the Nationalist camp.
Correct, the original leader was José Sanjurjo, but he died very early on in a plane crash. There's a whole Turtledove novel about him taking less luggage with him and surviving the flight, causing Spain to take less of an isolationist role in WWII.
Sanjurjo taking Spain into WWII is very unlikely, so Turtledove messed things up there. Under Sanjurjo, I think Spain would have been more isolationist and more akin to the Alfonsian monarchy than towards the crypto-fascist clerical authoritarianism of the
Estado Español of OTL. Thus, support for the other continental fascist powers would have been weakened by an inward-looking leadership.
Internal fighting on the Nationalist side could be one avenue to explore. There were a lot of varying factions and whilst the Republicans suffered infighting, the Nationalists somehow remained tentatively united.
This is one of the more important points when one looks at the Spanish Civil War. Franco's political effectiveness was in managing disparate factions with leaders that had huge egos, thus allowing the Nationalists to coalesce around Franco once Sanjurjo was dead.
A major PoD is José Antonio Primo de Rivera being rescued from prison in Alicante during the summer of 1936, which was actually an OTL plan. Having him take control of the Falange during the early days of the war would have seen a more strident fascist element in the Nationalist camp. Franco wouldn't have the propaganda of
el ausente to rally the fascists to his cause.
Emilio Mola living longer is another factor, as well as Gonzalo Queipo de Llano perhaps not declaring for the Nationalist side (or delaying his declaration), towards creating more dissent on the Nationalist side. Francisco Franco was
very lucky in our world - almost to the extent of being implausibly lucky.