Why are British cooking shows seemingly more lax in terms of cleanliness?

I would put this down to a number of factors - culture and location. The US is probably much more "Hygiene aware" due to the very high environmental temperatures found in some regions, unlike the UK, which is pretty temperate and generally a lot cooler. The US is also much more litigious than the UK, with extensive food safety warnings you would not normally find on UK/EU packaging. There is also a lot less fishing, hunting, shooting and homesteading culture in the UK, which would re-enforce core food safety values. I doubt if as many people here would be aware of botulism caused by garlic stored in oil for instance, or the discipline surrounding home canning. The lack of free universal healthcare in the US may also play a part, as a visit to the doctor is not cheap, and prevention being cheaper than cure. This all adds up to a greater sense of personal responsibility, whereas in the UK a stomach bug is casually dismissed almost as a rite of passage with a Gallic shrug (That dodgy pint of beer, kebab or curry after a night out etc.).

The "3 second rule" is pretty much a fact of life here rather than an urban myth, despite a somewhat biased study carried out by a floor mop manufacturer (https://www.dailymail.co.uk/femail/article-2138777/The-second-rule-fact-fiction-Scientists-reveal-food-dropped-floor-safe-eat.html). With a lot of people not cooking from scratch, and depending on take-away food or ready meals etc. this also prevents the adoption of best practices. I am considered totally OCD as I have a fridge, meat and gun thermometers I use when cooking. Not many people in the UK would be concerned with fridge temperatures, whereas in the US it is a much more critical issue etc.

The UK program probably just reflects these more "casual” values, I have seen both British and American chefs wear jewelry or rings on TV shows when handling food for instance, something that would be condemned in a professional environment either side of the pond.

Taking into account the temperatures and time required to record inside a television studio though, I wouldn't just be worried about the dropped food.

Not a complete answer, but I guess part of it may be that the bakers in GBB are amateurs, not professionals. Cooking shows where professional chefs, or chefs who wish to become professionals, compete hold to commercial safety standards. Amateurs, and those of us cooking and baking at home, can be a bit more relaxed (depending on your personal tolerance).