Were the Pullman coaches used on the British railways set up like they were in the US. Where the Pullman Company built,controlled and staffed their coaches. The railroad and Pullman had a sharing agreement on fares.
Not really the same because Pullman in the US was principally for sleeping cars, the two terms being somewhat synonymous (although not all US railroads used Pullman to manage their sleeping cars) whereas in Britain they were day cars, the UK sleepers were railway operated. Yes, ultimately the Pullman Co built (through independent builders), controlled and staffed the cars, which in the private railway days before 1948 could move around between companies. The former Southern Railway made the most use of them, and in fact contracted the staffing of their own catering vehicles out to Pullman. Pullmans provided both First (1+1) and Second (2+1) class accommodation like regular trains did. Regular train tickets were required, but in addition the Pullman conductor came round and charged everybody a supplementary fare. This didn't cover their total costs so there was an additional commercial agreement between railway and Pullman. Food was charged separately as well, served at seat rather than in separate dining cars. The Pullman headquarters was at Stewarts Lane in South London, alongside the locomotive depot there, where there were extensive works for maintenance and repair of the cars as well as one at Brighton, as well as the central commissary etc. I understand he railway paid for the cost of building and maintenance of the underframe and running gear, and Pullman paid for the body and interior. They did a lot of simplistic rebuilding and were some of the last wooden-bodied vehicles in main line service.
From about 1900 most of the shares were owned by the same company that owned Wagons-Lits across Europe, but there were various sales of shares and by the 1950s most ended up owned by British Railways, who finally bought out the remaining shareholders in the late 1950s. There was some rivalry between Pullman and BR catering staff on the same routes, which in part led to the separate company continuing, not least because they were members of different unions, and Pullman trains introduced in the 1960s and later were actually staffed by BR catering personnel, and Pullman was just a brand.