When I got Mr. Fish, his fins were in abysmal shape.
I took about 5 months for his fins to grow back fully and become fully re-colored. BUT he was missing almost his entire tail.
Depending how much of his fins are gone, you're looking at at least a month for full growth/recoloration. The color will slowly fade back into the fin.
My album documents some of the process, starting about two months in... this will show you how the clear fades to the coloration: http://www.bettafish.com/album.php?albumid=1328
This picture in particular is good for showing what regrowth looks like:
Note: This picture was taken after 3 months of growth. You can see where his fins were gone up to, if you look at the coloring in his fins... His tail was gone up to where the purple threads end... so about half of the size in this picture.
This was after about 4 months.