AnnLinda is correct, both parents must be KC registered., and the breeder is the only person who can register the puppy. The dam of the litter has to be registered in the breeder's name. You could start by contacting the Kennel Club 0870 6066750 with the name of the dam and her date of birth if known, and asking if she is registered, and also ask about the sire. They will tell you, and also whether any of the littter was registered.
It costs �12 per puppy to register the litter up to 12 months, but after that it costs �60 per puppy. If you contact the breeder and offer to pay, she may agree. If the dam is not registered you might still be able to get the breeder to do the same with her breeder, although it will be getting quite expensive by then.