I work in the music industry, I know about sync licensing.

But here's one anyway:
Music Clearance at ITV for independent production companies (doc)

That makes reference to karaoke performances and competitions as not necessarily being covered by ITV's own blanket PRS broadcasting licence, and every use will need to be cleared with ITV's music team regardless of this blanket licence.

I don't know whether FremantleMedia would classify as an independent production company, and it's possible they have their own licence as well directly with publishers, as the PRS music programmes blanket licence is only available to indies making content for Channel 4, E4, More4 and Channel 5. (PRS Music Programmes blanket licence)

In any event, whatever licence they have or don't have has nothing to do with the venue's music licence, as it's the network and production company that need licences for broadcasting.