No Sky involvement.
The Late Late Show is a Fulwell 74/CBS Studios Co-production.
The Tom Synder & Craig Ferguson incarnations were Worldwide Pants Inc. Productions, as that was part of the deal that got Letterman to CBS. It gave him control of the 11.35-01.35 programming.
When he (semi) retired, CBS Studios took more control of the 12.35 show.
My user name might look like Hatton Cross, but it's pronounced Throatwobbler Mangrove.