There are a whole bunch of reasons (physiological and psychological) why getting a cut like that produces a delayed sensation of pain. Psychologically, if you're deeply concentrated in something (e.g. immersed in listening to music), you might be subject to a form of delayed perception. Physiologically, the cut area may not send pain signals until an inflammatory response mounts against the bleeding. These could combine to produce a significant delay in your (conscious) perception of the wound, even without something like the AirPods.

(One could argue the degree to which it isolates you from the environment acoustically might also be insulating you from heeding your environment at all - sounds like something that might be worth studying!)