The argument discussed here was first published by Lucas in 1961 ("Minds, Machines and Goedel"). What seems to me a conclusive refutation of it was published by Hilary Putnam in 1960 ("Minds and Machines"). Putnam's point is much the same as simonh's in a comment here: the Lucas(/Penrose) argument makes assumptions about human mathematicians that do not in fact apply to human mathematicians.

[EDITED to add:] Contra simonh, though, one can refute the argument without going so far as to say that human mathematicians are definitely inconsistent. (Maybe a sufficiently careful human mathematician is consistent.) All that's required is that we not be able to prove that we are consistent, and I think it is extremely clear that we can't.

In other words, the Lucas(/Penrose) argument was refuted before it was ever published.

(Penrose's version isn't really any improvement on Lucas's.)

Note: The video is 2 hours long and highly technical. I haven't watched anything like all of it. The speaker is _not_ endorsing Lucas's or Penrose's conclusion that Goedel's theorem shows that minds cannot be mechanized; he makes observations similar to Putnam's, simonh's, and mine, but clearly makes them with more subtlety and intricacy :-).