The white colour and puffy texture of your skin is because of the bandaid trapping moisture and it not being able to evaporate from your skin. Every time I cut myself, which has happened a lot in the past, working with knives in a fast-paced kitchen environment, if the bandaid is left on for a day or more and is tightly wrapped, this will occur.
Disinfect with hydrogen peroxide, meaning dunk your thumb into a small cup of it, rinse with water at a temperature you can stand, pat dry and bandage loosely. After a period of a couple hours, let it air out by removing the bandage for a couple of hours each day until the wound has begun to scab and the tissues are visibly fusing back together.
At this point, the scabbing has provided an effective barrier against infection, and can be open to the air unless you're doing something where you may accidentally hit it etc.
I recently cut deeply into the back of my thumb across both knuckles and nicked a tendon. Though I would advise going to emergency if you lose any mobility as I had, nonetheless I closed and tended to the wound on my own and recovered.
Your wound does not sound very grave, I'm certain you can treat this yourself.
Though if there is a scent of vinegar to the wound, get yourself checked out. As mentioned before, gangrene is nasty,you don't want to lose your digit or the surrounding muscles.
