My ideas:
- For starters the German-Italian aid the Nationalist were getting would, not disappear, but dwindle some, as both countries would start to hoard resources for their attacks in Poland, France and North Africa.
- Considering that now the Republicans were supporting France and Britain in their fight against Fascism (nominally, at least), perhaps France wouldn't surrender so soon and Vichy France wouldn't be created: perhaps the French gov't would travel to Catalonia so that France stayed free, and from there sail to Northern Africa, where those forces that would have been serving Vichy in there instead helped in the fight against Italy, they might invade Libya in collaboration of Bernard Montgomery's 8th Army.
- The Lend-Lease Accord between USA and Britain would be expanded to African France and Republican Spain, with the latter being supplied from Malta and Argelia.
- Gibraltar is invaded by the Nationalists. This prompts Britain to declare war on National Spain.
- After the Fall of France (no Vichy France in here, the whole country would be under German occupation) Operation Barbarossa would be delayed a few months, maybe even a year, because Hitler would surely order an invasion of Republican Spain to help his ally Franco.
- Montgomery would have to work much more than in OTL, because his 8th Army would have to drive the Francoist army out of Morocco. Then, a new front could be opened starting from Gibraltar, making a Overlord-like operation but just crossing the Strait.
- Spain's pitiful prowess in their war against the Republic (we didn't exactly have a good army back then, and 4-5 years of continous war surely helps in the destruction of an army) makes Hitler divert resources from his war effort to help the Nationalists. Spain becomes something akin to Italy, sucking the resources of Germany. War would go easier for the Allies.
- After being invaded by Nazi Germany, Republican's Spain control is given to Franco. The Republic organizes a huge effort to transport as many troops as possible to French Algeria. These troops would then help in the liberation of Spain.
- By 1944, Spain would be in Allies hands. Overlord would happen in the same moment, but the "broad approach" tactic that was suggested in OTL is rejected in favour of a lightning attack to reach Paris. The Allies launch "Operation Roncesvalles", a three-pronged attack crossing the Pyrenees in the three points where a natural pass between Spain and France is placed: from the Basque Country/Navarre (aimed at Burdeos), Catalonia (towards Nice and Marseilles) and Aragon (directed to capture Toulouse).
- Sardinia and Corsica are invaded from the Balearic Islands.
- Soviets don't arrive to Vienna before the Allies do. This is because Roncesvalles obligues Hitler to send groups of the Wehrmacht to southern France, diminishing the number of soldiers in the Brenner Pass (I'm not sure if it's called like that) and allowing the Americans and British to arrive to the city.
- War in Europe might end before it does in OTL, because since Southern France has been liberated from Spain, the Army in Northern France and Italy doesn't have to send soldiers and tanks there.
- Germany wouldn't be divided in influence zones as it did in OTL, because, since Berlin fell into British-American hands, Stalin doesn't have the posibility to create the GDR.