Following the end of the Civil War, the Spanish Army adopted a newer and more modern uniform, along with a new helmet, similar but not quite identical to the German type. The older pattern of helmet was retained in some units until the 80's iirc.
The only illustration I have to hand is this one;

J & K would be typical soldiers for the 1940-50 period. La Legión wore a shirt of the same distinctive colour as during the Civil War.
L is a Compañía de Vanguardia member, which replaced the Asaltos as the urban police force.
M is a Carabinero, who were frontier/customs police.
The Guardia Civil's uniform remained unchanged iirc.
Spain went through a period of austerity and poverty after the Civil War, so in your scenario I'd expect Germany to provide uniforms and equipment in the same way as it did for the volunteers of the Blue Division.