Hi, Tracy. I don't think they "controlled" the appearance of the cars whose bodies were made by the famous coach builders you name. Rolls began building complete cars when the era when almost all Rolls Royces were chauffer-driven limousines was coming to an end, and owner-drivers started to become more common.
But the others continued to be custom made, with the customer essentially ordering from a narrow range of designs (sedan, limousine, sedanca, cabriolet, etc.), and an expansive list of trim and extras, like clocks, radios, cabinets, bars, folding seats and so on.
If you had money - and I mean money - you could order almost anything. An American woman specified a makeup case that cost some $60,000 in today's dollars and a special clock that cost $24,000. Others ordered duplicate instrument panels for the rear passengers in a limousine. One woman ordered a built-in chamber pot; even Rolls considered this a trifle eccentric but it was done. Others specified special interior heights for tall customers to be able to keep their top hats on, and one had all the operational switches labeled in Olde English script on ivory escutcheons. You could order leather or fabric seats, high-gloss woodwork or more satin-like. Cherry, mahogany, teak, etc.
The Maharajah of Patiala bought 35 Rollses in one order, specifiying gold-plated dashboards, real fur floor coverings and inbuilt medicine chests. The Nizam of Hyderabad, who could, at a pinch, come up with $40 billion (today's dollars), had 50, including one with a body made out of solid beaten silver.
The British army took several Silver Ghost chassis (frame, engine, gearbox and running gear) and fitted armoured car bodies to them. Lawrence of Arabia used one to shoot up the Turks. After the war he said that if he could have anything in the world it would be a Rolls-Royce with petrol and tyres to last his lifetime (which, sadly, was just a few more years.)
The so-called "Silver Lady" radiator mascot - correctly called "Spirit of Ecstacy" was sculpted by Charles Sykes, the model being the mistress of a British nobleman who had infused Rolls with needed cash when the company was finding its feet. In the days when radiator caps were removable, the owner was supplied with two; one with the Spirit on and a plain one, to be used if the car was to be left unattended for any time. When safety regulations came into effect, Spirit was hinged. When Queen Elizabeth II ordered a Phantom IV (a rare straight eight cyl.), instead of Spirit she specified a statue depicting St. George, England's patron saint, slaying the dragon. The first one was subjected to vibration on a testing machine, to simulate years of driving, and redesigned subtly to add strength.
Rolls began to lose interest in cars in the 60s, and concentrate on aero engines. Today they are the predominant manufacturer of jets, including the Trent, at 100,000 static thrust (5,000 used to be rated good in the 50s.)