There’s no warning-sign list just for the finger, so the general signs apply. Contact your provider right away for redness, increased pain, or yellow or green pus, or excessive clear fluid around the injury. These are signs of infection. Also contact your provider right away for black edges around the injury, which is a sign of dead tissue; bleeding that will not stop after 10 minutes of direct pressure; fever of 100°F (37.7°C) or higher for more than 4 hours; pain that will not go away even after taking pain medicine; and a wound that has come open or stitches or staples that have come out too soon.
Size sets the expectation on a finger. The smaller the wound, the quicker it will heal. The larger or deeper the wound, the longer it takes to heal. There is not much to heal in a small finger wound, which is what makes one that is not improving worth a second look rather than more patience.
How the wound happened matters as much as how it looks. Wounds include cuts, scrapes, puncture wounds, burns and pressure injuries. Not all wounds bleed. For example, burns, some puncture wounds, and pressure injuries do not bleed. A puncture from a splinter, a needle or a thorn can leave almost nothing to see on the surface.
Put that next to one more fact: deep wounds can affect tendons, muscles, ligaments, nerves, blood vessels, and bones. Those two facts pull in different directions, and the second is why a small finger wound that is not behaving like a small wound is worth showing to someone rather than reasoning about at home.
One more reason a finger wound may not settle: having a foreign body (such as a piece of wood or broken glass) in the wound can stop it from healing well. If something went in and you are not certain all of it came out, say so — that detail changes what gets looked for.