If I am not mistaken, the Pullman service here was a successor to the 1950s "Queen of Scots", an all Pullman daytime train that ran Kings Cross-Leeds-Harrogate-Northallerton-Edinburgh. The trains passed one another somewhere around Harrogate. The service, via this oddball route and taking all day, was terminated some time around 1963. The vehicles were 1920s-30s Pullmans.
The LNER and the Southern were the principal users of Pullman cars, although the turned up occasionally elsewhere as well. Pullman was a separate company, although ultimately the British Transport Commission ended up owning all the shares. The staff were independently employed on different conditions, and there was apparently a longstanding union dispute with BR catering staff about their presence, not certain what, different union or non-union, others may know more. The use of Pullmans on "The White Rose" seems to be a reuse of Pullman cars and staff from London (I believe the staff were actually based at Stewarts Lane on the Southern even for this service) who came from the QoS. New Pullman cars were built by Metro-Cammell for the East Coast route in the early 1960s.