A small knife cut usually takes a few days to start healing but after 4 or 5 hours the cut should fully close up and not bleed or need a bandage anymore.
Usually after 2 days the small cut from the knife should be healed at least somewhat but sometimes the small knife cut may take a few weeks to a month to fully heal.
Whenever I've cut myself a little with a knife or something else I can usually remove the bandage within a few hours and it has stopped bleeding and closed up.
Then I allow the wound to have some air which helps to speed up the healing process or you could put some Vaseline on the cut when it's not bleeding anymore which will keep the cut moist and can sometimes prevent a scab from forming.