I started taking doctor prescribed iron supplements. How long will it take for my blood to become normal?

Several months. Iron replacements for anemia may take several months to get your iron level back to normal. Too rapid a replacement may give you GI side effects like constipation or liver irritation. Take oral iron with vitamin c to help with absorption without a heavy meal.
Iron deficiency. Tough question. Many parameters to normal blood. It depends also how quickly you respond, how deficient you were, how well you absorb the iron, how immunocompromised you are, your overall diet and wellness status, and a repeat blood test.