A purge of the rebellious army officers and the gold shipments to Russia would have resulted in a weak, largely bankrupt government attempting desperate neutrality.
As several have already pointed out, the Iberian peninsula lacks riches, and Hitler's lebensraum was in the East, so a Nazi invasion seems unlikely. Yet a prostrate Spain might have been an attractive victim for Mussolini. The Fascist leaders were not reasonable men and logic does not apply.
Speaking of unreasonable, there's little doubt Churchill could have avoided scheming about the Spanish underbelly of Europe, just as with Greece and Italy - but hopefully overruled.
Imho, the only major factor might have been if German military power had been defeated or embarrassed on the peninsula, the early stages of WW II might have been very different.