I cought strep from my cousin but will it turn into scarlet fever like his did?

Maybe. If you really caught it from that cousin & not at random, the strain you have would be one that produces the red rash. Treatment started before the rash appears may stop it. Scarlet fever is really an archaic term left over from the pre-antibiotic era, when many fevered off & on for days with it.A common name applied to the rash now is scarlatina.
No way to know. Certain types of strep release a material that causes the scarlet fever rash. All scarlet fever is is strep throat with an associated fever and rash. The presence of the rash does not mean the infection is worse than strep throat without a rash. Treatment is the same in both cases: antibiotics. Make certain you take the medicine for the full course even if you feel better in a few days.