Here's what I managed to pick up by working at Trafficlink (who have a close relationship with ATOC and NRCC).
In the case of National Rail, it is split into two parts and administered by ATOC.
NRES (National Rail Enquiries Service) is the phone service for National Rail. They contract the service out to various companies with call centres in Plymouth, Newcastle and the Dearne Valley near Rotherham - the latter being in operation 24/7. There are another two call centres in India - one in Bangalore and the other in Mumbai... but those two might be a bit of a commute for you
I think BT manage the call centres in Newcastle, Plymouth and Bangalore, with the remaining two (Rotherham and Mumbai) being the ones managed by Ventura. I guess recruitment to be a call centre operative might be done by BT/Ventura and then allocated a company to answer phones for, rather than through National Rail themselves.
Contacting ATOC directly might be the best option.
NRCC (National Rail Communication Centre) are based in Doncaster and look after the National Rail website's 'Service Disruption' and 'Engineering Works' sections, as well as having Trafficlink bug them
By speaking to various sources they maintain those two sections of the website only, as well as a few non-public utilities.
I know nothing about thetrainline/QJump.co.uk etc I'm afraid
Might be worth speaking to 'CromptonKarl' about Raileurope though. Again I know nowt about them, but I think they do a similar thing to thetrainline.