Thank you for the query.
It is very hard to bleed to death from finger cut (injured blood vessels do contract so the bleeding stops). However it is possible if it were really a lot of blood.
, pale skin, cold
hands and feet, increased heart rate
(more than 100/min) are the symptoms of significant blood loss and should force your boyfriend to visit ER.
Moreover, the wound probable needs some stitches to heal property.
Hope this will help.