Those edits were done years ago when the channel was owned by Virgin and they're still using them now.
As Bullseye was made by Central you'd go to ITV for that back catalogue (of what they own anyway).
As a general rule: stuff made by an ITV company you'd go to ITV Studios for (for stuff they actually own or have inherited, which is most of the network output from England bar Thames (go to Fremantle), TVS (good luck with that!) and a few other things), shows the BBC made themselves you go to the BBC (obviously
) and indie stuff you either go to whoever made it (if they still exist) or whoever's bought the rights to that company's catalogue. This includes things like Total Wipeout which was made for the BBC but was actually IIRC Endemol. Crystal Maze, Interceptor and Treasure Hunt were all made for Channel 4 by Chatsworth but Chatsworth hasn't existed for years since Malcolm Heyworth retired but the rights live on somewhere since Challenge let the Maze expire and they bought them back again so somebody's obviously managing those assets.
As to what they receive, I believe its possibly a transmission master, ie what was intended to be broadcast at the time, with "call this number or write to this address to enter this competition" sort of references, its probably Challenge's responsibility to take all that out before it airs, otherwise people tend to be too thick to realise they're watching a 20 year old programme...