One gets the impression that the British and French governments would have sided openly with Franco if they weren't worried about their electorates.
However, if a few battles go the Republic's way and Franco still hasn't sealed the deal by September 1939, there are a number of implications.
1. This would likely be a negotiating point with the Nazi-Soviet Pact. The likeliest point of agreement would be that both sides withdraw support from their respective factions. Generally, Spain would be put in the Nazi sphere, but Stalin would likely extract the concession of no more German aid for Franco, and Hitler would agree since those resources would be needed for his war.
2. The British and the French continuing to let Germany ship aid to its allies in Spain would be a bit ridiculous once they are war with Germany.
3. Italy, as a neutral country and not a party to the Nazi-Soviet Pact, could continue to expand its role in Spain and probably would. But this might leave them less prepared to fight Britain.
4. Things get interesting after the fall of France. If Franco still hasn't defeated the Republic, his meeting with Hitler goes very differently. Hitler could offer help, including army and air force, to defeat the remaining republicans, and make it clear that he would do the job anyway if Franco refused the help. There would be no questions of concessions to Spain. Franco would not have to declare war on Britain, but would have to let the Nazis take Gibraltar. With Churchill in power, this likely results in a British declaration of war on Spain anyway as Franco doubtless new. Also, if Franco didn't agree he could be replaced by a real fascist dictator.
5. Increased Italian military efforts in Spain will probably butterfly the invasion of Greece. With Spain to play in, Mussolini doesn't need the prestige and even he should realize he can't do both. Probably by April 1941 the Italian and German forces are mopping up in Spain and securing Gibraltar. Unlike IOTL, the republican forces take to guerilla activity and this time there is the Special Operations Executive in London to support them.
6. The Nazis taking Gibraltar also starts having butterflies.
7. Iberia gets to be a theater of World War 2 if for some reason the Axis attacks Portugal. Churchill would probably love to see this happen and would do what he can to make it happen.
8. If Hitler avoids getting further involved in Iberia, such as attacking Portugal, he benefits on the whole by holding Gibraltar and no Balkan campaign, I'm not sure if enough to make a difference.
9. IOTL, the Spanish shipped minerals to Germany, troops to the Eastern Front, and provided help for the u-boats, so I don't know if Spain could really do more than this. Probably not enough to justify the increased economic assistance.
10. One way or another, the North African campaign is seriously downgraded. The logistics are still impossible for Rommel, but with no Mediterranean convoys, or worse, an active theater in Iberia, the British are only going to do a holding effort. This also means Rommel can't take supplies from his overexposed opponents, so that cancels out his own convoys getting through easier. The Afrika Korps may not even be formed in this timeline. Malta is toast however. Also there is not going to be Torch.