Sky Chrome Media might be able to claim a copyright on their transfer, not the show itself. (Restored video transfers of PD films have been bootlegged by unscrupulous companies before.) If EON Productions really wanted to, they could claim copyright on the character of Bond in a effort to discourage sales.
The VHS cover in the first post used the 007 gun logo, which I'm pretty sure is copyrighted.
Either EON or MGM requested the Neil Connery spoof "Operation Double 007" be retitled for the recent MST3K episode release. The title card as seen on the show wasn't changed, but the DVD packaging reflects the original film's title "Operation Kid Brother". The title card had apparently been altered by whoever owned the movie in the 90's when MST got their hands on it.
A legal claim might not hold water, but to fight it out in court is expensive.