You obviously missed the beginning of the programme. John Sergeant was quite specific that Westland had taken the Sikorsky airframe and reworked it completely, starting with the Rolls Royce Gnomes, " and you don`t get much more British than that" were his words.
Many old Sea King lags went on to spend many a happy hour ( and some not so happy) in the S61 and it was a wonderful aircraft, some would say yet to be improved on in spite of the march of technology, but it was not a Sea King.