This is a good contender for a copyright violation either directly or as a derivative work.
If certain elements of the show are so well-recognized in the British public or subgroups of it that if imitated or replicated would cause a reasonable person to recognize the original show, the owners of the show could potentially apply for a trademark or sue in certain conditions, but less plausible as on copyright groubds.
Business methods are generally not more than abstract ideas. A good analogy is a computer algorithm. One can draw a flow chart or could theoretically otherwise carry out the operations of the algorithm in their head, for example calculating the area of a right-angle triangle, but not even Pythagoras could assert the mathematical method was an invention as the concept excludes intangible things, it is a discovery no doubt if novel and non-obvious. A machine somehow analogously calculating the result would be an invention. Similarly, a business method can be imagine as an algorithm and it runs in people. If it’s merely people necessary there is no patent eligible subject matter in the method. In a time where there were no vending machines, the method to provide a can of coke upon the registering of sufficient payment could be the subject of a patent as a system and separately as a method carried out on the machine. One can reasonably refer to such a method as a business method as the process results in direct revenue.
It is doubtful that the referenced examples would not be rejected as lacking patent eligible subject matter, that is, it be more than an abstract idea, and even if some extremely narrow claim may be granted, it would not have commercial value as it could be easily worked around.
Whether or not they would operate until anyone tells them to stop would require speculation, but whether or not they then could be compelled to stop is a question on-topic here, and the answer is they may very well be compelled to stop if discovered or the IP holder decides to do so.