How much time does it take to regrow hair after an increase in iron levels?

If
you truly have iron deficiency as the cause of your hair loss, you need to
treat your condition.  Your hair growth
may begin shortly thereafter, however it will take at least 6 months to see the
volume of hair recover back to normal. 

For many with hair loss, low iron levels has nothing really to do with their hair loss. Ferritin levels come up and hair density does not change. However, if you are in fact an individual in who iron is a causative factor in your hair loss, it will take 3-6 months for an improvement. As a hair specialist, I would love it if iron had more of a role for my patients, but far too much emphasis is placed on iron levels in hair loss which ultimately leads to disappointment for many when hair does not improve after appropriate supplementation.
REFERENCE Sinclair R. There is no clear association between low serum ferritin and chronic diffuse telogen hair loss. Br J Dermatol. 2002 Nov;147(5):982-

Hair regrowth after restoring your Iron levels can take up to a year or more. Some people, if they also have the genes for hair loss, may not get the hair to return