I'm not aware that any organisation limits the number of breeders. The GCCF registers pedigree cats and provides guidelines to breeders as to how they operate (things like not rehoming kittens until 13 weeks old, fully vaccinated etc) and can suspend breeders for bad practice (eg selling unhealthy kittens), but doesn't impose any limits on the number of breeders. I forget where I read it (so dont' know whether it was a general guideline or what) but I'm sure I read somewhere that breeders should take note of market forces and if they struggle to sell their kittens (eg if a lot of breeders of that breed in the same area) then they should stop breeding or breed a different breed. I have a feeling it wasn't on the GCCF website though. Maybe on a cat club website. I think the only regulation of breeders/breeds is that there are certain breeds they will not register, eg Scottish Fold, Munchkin and although Bengals are registered, they have said they will not register any further breeds created by using wild cats.
The GCCF is the only UK organisation that exists, but FIFe and TICA are international organisations that have a UK branch, so to speak. They are both relative newcomers in this country and, as such, the number of cats registered with either of them is substantially smaller than numbers registered with the GCCF. FIFe is the Fédération Internationale Féline, and the UK branch is Felis Britannica. The latter was only established a few years ago (2003). TICA is The International Cat Association which now organises shows in the UK (since 2004). I don't know how they differ in terms of registering cats or in providing legislation and guidance for breeders, but their shows are operated differently.
I'm not entirely sure about what actual regulations the GCCF have. I know that the Somali cat club introduced PK deficiency testing as compulsory (meaning that members of the somali cat club abided to undergo testing on all cats used for breeding) and have been recenly been successful in getting the GCCF to accept their breeding and registration policy, which I think means that all somalis registered with the GCCF must have a certificate showing them to be PK deficiencey negative. So I'd guess there are other similar guidelines for other breeds.
Various cat clubs are affiliated to the GCCF, and cat shows run under GCCF rules are held by the clubs themselves and not the GCCF. The only show actually held by the GCCF is the Supreme. TICA and FiFE also hold cat shows, but there are fewer than there are GCCF shows.
Witty's your person to ask about this as she knows more about the world of showing and knows breeders. I think Witty shows mainly GCCF, but you could also try contacting Lottie as she shows with TICA or FIFe (I forget which) and Linda (Bengalbabe) who shows with TICA or FIFe also.