Poor Lonely Computer: Prince's Misunderstood Relationship with the Internet
npr.org
npr.org
I'm not sure I understand what the actual basis of copyright is in a situation like this.
I believe the videographer would hold a copyright claim to the original video recording, the artist Prince would claim copyright to his own rendition of the song and that specific performance, and Radiohead, Thom Yorke, or Radiohead's record label would collectively hold mechanical and phonorecord copyrights to the original song according to the song's history and their agreements.
It's trivial for Prince to obtain a compulsory mechanical license to perform a cover which would have been done prior to performance. It's also true that Radiohead may have the ability to prevent the video's distribution, depending on the interpretation of copyright law, and I find it also reasonable that Prince would have a copyright claim to the performance that he or his record label could have it removed. I certainly don't think Thom Yorke could supersede Prince's request to have the recording removed, just because he wrote or performed the song originally.
There is some information here, but it's still limited in scope: https://en.wikipedia.org/wiki/Compulsory_license#United_Stat...
I wouldn't believe what's written on issues of copyright law in offhand observations.
(Obviously, in most cases there are actual agreements hammering out various angles, like recorders or performers actually being configured as "work for hire" in order to take them out of the copyright pipeline altogether. But that's another story.)