If it is in the public domain does this override the Standard Youtube Licence? I.e. can on legally download the video? Or does one still have to use the AP's webpage which seems to require sign up (and then I don't know what).
If the content that's on Youtube is in the public domain then yes. Note that sometimes restoration involves enough creative work to make the result copyrightable even if the original footage was in the public domain.
I don't think you can legally download the video from Youtube; their TOS prohibits this. Of course I don't think Youtube has sued anyone for this and it's not a big deal.
Copyright wise it's legal I believe to use the content.