How quickly does cataracts develop in the eye do to eye injury?

Varies. From immediately to years after the injury.
Trauma & Cataracts. Cataracts (white lens change): most often due to or made worse by (most common first): aging, genetics predisposition, excessive sun exposure, smoking, diabetes, steroids, previous trauma, inflammation, infection in the eye causing inflammation; previous intraocular eye surgery; other genetic issues; can be congenital. Cataract can occur at time of trauma or take yrs to occur depends on injury.