If Nationalist Spain had joined the Axis, as was planned, then it would have had a tremendous effect on World War II. Gibraltar would have been taken from Great Britain, cutting off the most important entrance to the Med Sea. This would have left Malta and North Africa cut off from immediate supply, which would have forced all Allied shipping past Aden and through the Suez Canal. More importantly, however, it would have allowed German surface raiders into the Med, which would have been able to bolster Axis naval power in that sea while reducing the Royal Navy. This would have left North Africa, and the Suez cut off and most likely conquered. This would have secured Italy and all it's gains in that theatre.
The Nationalist forces while exhausted were mobilised in 1939, and would have been able to bolster the Axis numbers but only in the infantry department at first. Militarily they'd have been little use, unless used in garrison duties.
On top of this, the route out of Axis Europe by many Allied airmen would have been cut-off as German forces would have most likely been present in Spain. Making the usual escape to Gibraltar, then home, much more hazardous.
Spanish forces would have helped occupy Morocco, as there was a Spanish Morocco anyway. And this would have hampered Anglo-American efforts if an operation like 'Torch' were to take place in the 'alternative war'.
Spain did make a few actions during the initial stages of World War II. Taking advantage of the German invasion of France, Nationalist Spain captured the international zone, Tangier on the north Moroccan coast.