I don't know the proper term/phrase for this for this but does regular non News BBC programming from the English regions flow from the BBC One playout through the regions presentation gallery, back to the CCR (is that where the MUXes are coded) and then to the transmitter?
Not through the gallery, but through the building (apart from Jersey - which I think has some kit in Plymouth to handle the sub-opt). These days the BBC English regions are permanently fed a feed of BBC One Network sustaining (previously they were fed the BBC One London / London and South East version) which is routed through an SDI opt-switch, and then the output of this switch is fed back to Central Coding and Mux to be compressed and encoded, and then passed on to the terrestrial transmitters, satellite uplinks and cable headends (I think). There are dual paths for the entirety of this process.
The opt-switch in each English region switches between the incoming network feed and the output of the local studio. If this switch is used to switch from network to a local studio source, then this is described as a hard (or in some cases - crash) opt. However it is often the case that the studio will itself have the incoming network feed as a source, and the opt-switch switching from network incoming to network via the local gallery is called as soft opt. (As the opt-switch happens before the junction the audience sees - which is then done purely in the gallery)
This is how things currently work. At one point the encoding for both terrestrial and satellite broadcasts took place in each regional centre, rather than centrally, which meant BBC One couldn't be statmuxed in England, and also that the full PSB1/BBC A mux passed through the regional centres. There were also all kinds of delays and tally-driven opt-switches to allow opts on both digital and analogue networks (where digital was late compared to analogue)... Luckily long gone.
Or do they just feed the opts to CCR (doesn't ITV follow this method) ?
ITV work by sending studio outputs to their network playout areas. Arguably this allows for greater resilience, but it does remove some independence...