In X-Men: First Class, in Nazi Germany, Dr. Schmidt is a human. Later, during the Cuban missile crisis, when he goes by Sebastian Shaw, he is a mutant with powers. How did he become a mutant?
I don't see why you'd think Shaw was a mere human in Nazi Germany. Sure, he might not have made a show of his powers, but nothing indicates he was human; in fact, there are two scenes that very much indicate the contrary.
The first is when Shaw absorbs a grenade.
I've got the power to absorb energy. Keeps me young.
Presumably he has been doing this for some time. Even before the 40s, probably.
The second is a combination of what he explains to young Erik in the beginning, about evolution and a new age in the human reign, and the dialogue in the final confrontation, when Magneto explains how Shaw was unlocking his powers. This very much indicates that Shaw was a mutant all along, IMO.