This happened to me recently so I looked (post facto) for the answer.
R David Sperling gives background here
The Shulchan Aruch (Yoreh De'ah 66:10) states that human blood, after
it has left the body, is forbidden. This is not because the human
blood itself is forbidden to us from the Torah, but rather because
someone might think mistakenly that it was non-human blood and
therefore forbidden (marit aiyin).
one bites an apple and finds that blood has come out of one's gums
onto the apple, the blood spots must be removed from the apple before
taking the next bite. However, continues the Shulchan Aruch, blood
inside one's mouth is allowed, and so if one has bit their cheek, or
has bleeding gums, the blood inside the mouth may be swallowed, and
one does not need to spit it out.
There are two opinions regarding blood from a wound:
- Some (following Tosafot, e.g., Orach Mishor, Kaf Hachayim), say that it is permissible to suck it as it is clearly human blood that has not separated from the body
- Others (following Rashi, e.g., Nachal Eshkol, Minchat Yaakov) rule strictly that only blood inside the mouth which others cannot see can be swallowed
For more details and sources see here from Eretz Chemda and Nishmat Avraham vol. 2, p. 16.
Of course, consult your rabbi
before implementing anything you learn here. Mine ruled strictly that one shouldn't suck it once it has left the wound and is visible to others.