You do not have necrotising fascitis or any other kind of infection in your finger because if you did you would know big time - my son 7 yrs ago stood on something sharp with bare foot and pricked it - so small not visible with the naked eye and these are the dangerous ones as they don't bleed to clean the wound. Within 24 hrs he had a black foot and red lines up both side of his leg to knee and was seriously ill. It took a week in hospital and iv antibtiotics and at first they were worried about him losing his foot! He made a full recovery in a week. This was caused by Strep bacteria on his foot getting into a pinprick wound. I was told that bleeding is cleansing. Because he didn't really notice the pin prick he didn't clean his foot and wore sweaty socks all day. If he had washed his foot or applied any sort of disinfectant he would not have got that infection.

Your cut sounds perfect for a cut