Cuts on the inside of the nose are problematic. They're in an enclosed space with no light and little air. Worse, your sinus system keeps sending out mucous full of bacteria and other infectious nasties to flow right over the cuts. They're hard to reach, and a band-aid won't stick. However, with proper treatment you can get them healed up much as you would any other small cut or scratch.
Mix 1/2 tsp. of salt in a cup of lukewarm water. Stir thoroughly.
Tilt your head back and pour 1 tsp. of the mix into the cut nostril.
Blow your nose vigorously into the paper towels. Use paper towels instead of tissue paper. You'll be surprised how much comes out.
Coat the cotton swab with antibacterial cream. Swab it gently on the cut section of your nostril.
Apply this treatment once per day. Between treatments avoid as much as possible blowing or picking your nose, as these can aggravate the cut.