Paul Merton I know has said a number of times that Have I Got News For You involves very little preparation outside the role of host often saying that it’s a show about the news, reading the paper is the only prep needed! The only thing he says is shown to the panel prior to filming are the pictures for the caption competition and I’ve heard that from a few fellow panellists as well.
I've heard Merton say that after a while he didn't even bother reading the papers. He also said early on that some guests used to come in with little routines they'd scripted and ask if Paul could do them with them, which he always refused to do. It would be pretty obvious if some of the non-comedy guests were reading scripted lines, sometimes the jokes die on their arse when proper comedians do them, never mind someone who isn't a comedian. It would be blatantly obvious.
But obviously if you're going on a panel show you'd think of jokes you might like to use, same as when you're going on a chat show you'd think of what stories you wanted to tell. Most comedians will also have a mental shelf of jokes they can use in various circumstances. It works out OK if it seems natural. Jason Manford used to be a bugger for unsubtly shoehorning bits of his stand-up in, often with the most tenuous link to the subject, he never managed to make it look like he was improvising.
As for that Kevin Bridges clip, brilliant though it is, there are some obvious bits in it where he's clearly told the story before and refined it ("two Scottish guys trying to speak English", and so on), but that's fine as long as it's funny, which it is. It shouldn't matter if they prepare stuff, they don't win anything so it's hardly cheating. It's what they should do, really.
Though there was a time at the end of the original UK version when it was the same few American performers doing essentially the same games every week. Helping Hands was one of the repetitive rounds, it turned into an excuse just to get Ryan Stiles just to eat horrible stuff
This is a bit of a Dan Patterson trope, there is the tendency when you have very reliable participants to want them on as much as possible, so when Ryan Stiles became massively popular they hired him every week, and because he worked so well with Colin Mochrie he was on every week, and eventually you get a load of regulars and it's the same every week. It's not so much on Mock The Week now but a few years ago that was absolutely stuffed with regulars as well. It's alright if you enjoy watching them, and they can come up with the goods, but they are massive eaters of material so the show ends up getting a bit repetitive.