I asked someone who has been collaborating with Dan Kan recently to ask him who first discovered the nerve, and I'll just paste the relevant part of the e-mail below:
In other news, I asked Dan Kan who really invented the nerve. He didn't know precisely, but after we talked about it for some time, we agreed that Mac Lane's work on the classifying space of a group suggests that he must have been one of the first — if not the first — to have discovered the nerve functor.
I'm going to leave this unattributed for now, since I haven't asked for permission to post it, but for the same reason, I'm also making it community wiki.
As for why Dan Kan would be a reputable source for this issue, read my comments on Andy's post.
Maybe Dan Kan (or my once-removed source) will appear on MO to flesh this answer out!
($\uparrow$ not foreshadowing!)
Anyway, this answer seems to support Sean's original conclusion.