As the Republican forces were defeated and I doubt Franco would want hostile foreigners to remain in the country, staying in Spain was probably not an option. Unless they wanted to join the forces that continued the fight as an insurgency that is.
I know that some went into exile in the USA, Scandinavia or the USSR, but surely some of these former combatants attempted to return home. I imagine they would be subject to arrest and interrogation but I'm having trouble finding any sources that speak about it. What about those from countries that would soon be annexed or occupied by the Axis? Thousands of French, Czech, Dutch and Polish citizens (among others) also went to Spain and I imagine they would be more likely to return home than their German and Italian former comrades.
My older answer here covers a lot of the ground you're interested in, but if you want to follow up any of the points I can address them here!